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, reflected to the red mirror, and then reflected back to the blue mirror. Find the distance that the light travels from the red mirror back to the blue mirror. PT Blue mirror 4. 7 f t 25° O 6 ft 2. A triathlete sets a course to swim S 3 4 E from a point on shore to a buoy mile away. After swimming 300 yards through a strong current, the triathlete is off course at a bearing of S E. Find the bearing and distance the triathlete needs to swim to correct her course. 35 25 35° 25° 300 yd mi 3 4 Buoy W N S E 3. A hiking party is lost in a national park. Two ranger stations have received an emergency SOS signal from the party. Station B is 75 miles due east of station A. The bearing from E and the bearing from station station A to the signal is S B to the signal is S W. 60 75 (a) Draw a diagram that gives a visual representation of the problem. (b) Find the distance from each station to the SOS signal. (c) A rescue party is in the park 20 miles from station A at a bearing of S E. Find the distance and the bearing the rescue party must travel to reach the lost hiking party. 80 4. You are seeding a triangular courtyard. One side of the courtyard is 52 feet long and another side is 46 feet long. The angle opposite the 52-foot side is 65. (a) Draw a diagram that gives a visual representation of the problem. (b) How long is the third side of the courtyard? (c) One bag of grass covers an area of 50 square feet. How many bags of grass will you need to cover the courtyard? (i) (v) (ii) (iv) (iii) v v v 5. For each pair of vectors, find the following. u v u v u v u 0, 1 v 3, 3 u 2, 4 v 5, 5 u u u u 1, 1 v 1, 2 u 1, 1 2 v 2, 3 (vi) (b) (d) (c) (a) 6. A skydiver is falling at a constant downward velocity of 120 miles per hour. In the figure, vector u represents the skydiver’s velocity. A steady breeze pushes the skydiver to the east at 40 miles per hour. Vector v represents the wind velocity. Up 140 120 100 80 60 40 20 u |
v W −20 Down 20 40 60 E (a) Write the vectors and s u v. (b) Let u v in component form. Use the figure to sketch To print an enlarged copy of the graph, go to the website, www.mathgraphs.com. s. s. (c) Find the magnitude of What information does the magnitude give you about the skydiver’s fall? (d) If there were no wind, the skydiver would fall in a path perpendicular to the ground. At what angle to the ground is the path of the skydiver when the skydiver is affected by the 40 mile per hour wind from due west? (e) The skydiver is blown to the west at 30 miles per hour. Draw a new figure that gives a visual representation of the problem and find the skydiver’s new velocity. 493 333202_060R.qxd 12/5/05 10:48 AM Page 494 7. Write the vector terminal point of w w in terms of and bisects the line segment (see figure). given that the v, u v w u u is orthogonal to v and w, then u is 8. Prove that if orthogonal to cv dw for any scalars and c d (see figure). When taking off, a pilot must decide how much of the thrust to apply to each component. The more the thrust is applied to the horizontal component, the faster the airplane will gain speed. The more the thrust is applied to the vertical component, the quicker the airplane will climb. Thrust Lift Climb angle θ Velocity FIGURE FOR 10 θ Weight Drag v w u (a) Complete the table for an airplane that has a speed of v 100 miles per hour. 0.5 1.0 1.5 2.0 2.5 3.0 v sin v cos (b) Does an airplane’s speed equal the sum of the vertical and horizontal components of its velocity? If not, how could you find the speed of an airplane whose velocity components were known? (c) Use the result of part (b) to find the speed of an airplane with the given velocity components. (i) (ii) v sin 5.235 v cos 149.909 v sin 10.463 v cos 149.634 miles per hour miles per hour miles per hour miles per hour F2 F1 and and 9. Two forces of the same magnitude act at |
angles respectively. Use a diagram to compare the work 2, F1 in moving along the PQ if 2 60 and 1 done by with the work done by vector 1 1 30. (b) F2 2 (a) 10. Four basic forces are in action during flight: weight, lift, thrust, and drag. To fly through the air, an object must overcome its own weight. To do this, it must create an upward force called lift. To generate lift, a forward motion called thrust is needed. The thrust must be great enough to overcome air resistance, which is called drag. For a commercial jet aircraft, a quick climb is important to maximize efficiency, because the performance of an aircraft at high altitudes is enhanced. In addition, it is necessary to clear obstacles such as buildings and mountains and reduce noise in residential areas. In the diagram, the angle is called the climb angle. The velocity of the plane can be represented by a vector with a vertical (called climb speed) and a horizontal component is the speed of the plane. where component v sin v cos, v v 494 333202_0700.qxd 12/5/05 9:38 AM Page 495 Systems of Equations and Inequalities 7.1 7.2 Linear and Nonlinear Systems of Equations Two-Variable Linear Systems 7.3 Multivariable Linear Systems 7.4 7.5 7.6 Partial Fractions Systems of Inequalities Linear Programming 77 Systems of equations can be used to determine the combinations of scoring plays for different sports, such as football AT I O N S Systems of equations and inequalities have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Break-Even Analysis, Exercises 61–64, page 504 • Sports, • Data Analysis: Prescription Drugs, Exercise 51, page 529 Exercise 77, page 550 • Data Analysis: Renewable Energy, • Electrical Network, Exercise 71, page 505 Exercise 65, page 530 • Acid Mixture, Exercise 51, page 516 • Thermodynamics, Exercise 57, page 540 • Investment Portfolio, Exercises 47 and 48, page 561 • Supply and Demand, Exercises 75 and 76, page 565 495 333202_0701.qxd 12/5/05 9:39 AM Page 496 496 Chapter 7 Systems of Equations and Inequalities 7.1 Linear and Nonlinear Systems of Equations What you should learn • Use the |
method of substitution to solve systems of linear equations in two variables. • Use the method of substitution to solve systems of nonlinear equations in two variables. • Use a graphical approach to solve systems of equations in two variables. • Use systems of equations to model and solve real-life problems. Why you should learn it Graphs of systems of equations help you solve real-life problems. For instance, in Exercise 71 on page 505, you can use the graph of a system of equations to approximate when the consumption of wind energy exceeded the consumption of solar energy. The Method of Substitution Up to this point in the text, most problems have involved either a function of one variable or a single equation in two variables. However, many problems in science, business, and engineering involve two or more equations in two or more variables. To solve such problems, you need to find solutions of a system of equations. Here is an example of a system of two equations in two unknowns. 2x y 5 3x 2y 4 Equation 1 Equation 2 A solution of this system is an ordered pair that satisfies each equation in the system. Finding the set of all solutions is called solving the system of equations. is a solution of this system. To check this, you For instance, the ordered pair can substitute 2 for and 1 for in each equation. 2, 1 y x Check (2, 1) in Equation 1 and Equation 2: 2x y 5 22 1? 5 4 1 5 3x 2y 4 32 21? 4 6 2 4 Write Equation 1. x Substitute 2 for and 1 for Solution checks in Equation 1. ✓ y. Write Equation 2. x Substitute 2 for and 1 for Solution checks in Equation 2. ✓ y. In this chapter, you will study four ways to solve systems of equations, beginning with the method of substitution. Method Section Type of System 1. Substitution 2. Graphical method 3. Elimination 4. Gaussian elimination 7.1 7.1 7.2 7.3 Linear or nonlinear, two variables Linear or nonlinear, two variables Linear, two variables Linear, three or more variables © ML Sinibaldi /Corbis Method of Substitution 1. Solve one of the equations for one variable in terms of the other. 2. Substitute the expression found in Step 1 into the other equation to obtain an equation in one variable. 3. Solve the equation obtained in Step 2. 4. Back-substitute the |
value obtained in Step 3 into the expression obtained in Step 1 to find the value of the other variable. 5. Check that the solution satisfies each of the original equations. The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. 333202_0701.qxd 12/5/05 9:39 AM Page 497 y2 x 2 4 x Exploration Use a graphing utility to graph y1 in and the same viewing window. Use the zoom and trace features to find the coordinates of the point of intersection. What is the relationship between the point of intersection and the solution found in Example 1? Section 7.1 Linear and Nonlinear Systems of Equations 497 Example 1 Solving a System of Equations by Substitution Solve the system of equations. x y 4 x y 2 Solution Begin by solving for y 4 x Equation 1 Equation 2 y in Equation 1. Solve for y in Equation 1. Next, substitute this expression for single-variable equation for x. y into Equation 2 and solve the resulting 2x 6 x 3 Write Equation 2. Substitute 4 x for y. Distributive Property Combine like terms. Divide each side by 2. Finally, you can solve for y 4 x, to obtain y by back-substituting x 3 into the equation The solution is the ordered pair You can check this solution as follows. Write revised Equation 1. Substitute 3 for x. Solve for y. 3, 1. into Equation 1: Write Equation 1. Substitute for x and y. Solution checks in Equation 1. ✓ into Equation 2: Write Equation 2. Substitute for x and y. Solution checks in Equation 2. Check Substitute Substitute 3, 1 Because of equations. satisfies both equations in the system, it is a solution of the system Now try Exercise 5. The term back-substitution implies that you work backwards. First you solve for one of the variables, and then you substitute that value back into one of the equations in the system to find the value of the other variable. Because many steps are required to solve a system of equations, it is very easy to make errors in arithmetic. So, you should always check your solution by substituting it into each equation in the original system. 333202_0701.qxd 12/5/05 9:39 AM Page 498 498 Chapter 7 Systems |
of Equations and Inequalities Example 2 Solving a System by Substitution A total of $12,000 is invested in two funds paying 5% and 3% simple interest. r where (Recall that the formula for simple interest is is the time.) The yearly interest is $500. How is the annual interest rate, and much is invested at each rate? is the principal, I Prt, P t Solution Verbal Model: 5% fund 3% fund Total investment 5% interest 3% interest Total interest x y When using the method of substitution, it does not matter which variable you choose to solve for first. Whether you solve for first, you first or will obtain the same solution. When making your choice, you should choose the variable and equation that are easier to work with. For instance, in Example x in Equation 1 2, solving for is easier than solving for in Equation 2. x Te c h n o l o g y One way to check the answers you obtain in this section is to use a graphing utility. For instance, enter the two equations in Example 2 y1 y2 12,000 x 500 0.05x 0.03 and find an appropriate viewing window that shows where the two lines intersect. Then use the intersect feature or the zoom and trace features to find the point of intersection. Does this point agree with the solution obtained at the right? Labels: System: x 0.05x y 0.03y 12,000 Amount in 5% fund Interest for 5% fund Amount in 3% fund Interest for 3% fund Total investment Total interest x 0.05x 500 y 0.03y 12,000 500 (dollars) (dollars) (dollars) (dollars) (dollars) (dollars) Equation 1 Equation 2 To begin, it is convenient to multiply each side of Equation 2 by 100. This eliminates the need to work with decimals. 1000.05x 0.03y 100500 Multiply each side by 100. 5x 3y 50,000 Revised Equation 2 To solve this system, you can solve for x in Equation 1. x 12,000 y Revised Equation 1 Then, substitute this expression for resulting equation for y. x into revised Equation 2 and solve the 5x 3y 50,000 512,000 y 3y 50,000 60,000 5y 3y 50,000 2y 10,000 y |
5000 Write revised Equation 2. Substitute 12,000 y for x. Distributive Property Combine like terms. Divide each side by 2. Next, back-substitute the value y 5000 to solve for x. x 12,000 y x 12,000 5000 x 7000 Write revised Equation 1. Substitute 5000 for y. Simplify. The solution is at 3%. Check this in the original system. 7000, 5000. So, $7000 is invested at 5% and $5000 is invested Now try Exercise 19. 333202_0701.qxd 12/5/05 9:39 AM Page 499 Exploration Use a graphing utility to graph the two equations in Example 3 x2 4x 7 2x 1 y1 y2 in the same viewing window. How many solutions do you think this system has? Repeat this experiment for the equations in Example 4. How many solutions does this system have? Explain your reasoning. Section 7.1 Linear and Nonlinear Systems of Equations 499 Nonlinear Systems of Equations The equations in Examples 1 and 2 are linear. The method of substitution can also be used to solve systems in which one or both of the equations are nonlinear. Example 3 Substitution: Two-Solution Case Solve the system of equations. x2 4x y 2x y 7 1 Equation 1 Equation 2 Solution Begin by solving for expression for y y in Equation 2 to obtain into Equation 1 and solve for y 2x 1. x. Next, substitute this x 2 4x 2x 1 7 x 2 2x 1 7 x 2 2x 8 0 x 4x 2 0 x 4, 2 Substitute 2 x 1 for y into Equation 1. Simplify. Write in general form. Factor. Solve for x. Back-substituting these values of 4, 7 produces the solutions x and y to solve for the corresponding values of 2, 5. Check these in the original system. Now try Exercise 25. When using the method of substitution, you may encounter an equation that has no solution, as shown in Example 4. Example 4 Substitution: No-Real-Solution Case Solve the system of equations. x y 4 x2 y 3 Equation 1 Equation 2 Solution Begin by solving for expression for y y y x 4. in Equation 1 to obtain x. into Equation 2 and solve for Next, substitute this Substitute x 4 for y into Equation 2. Simplify. Use the Quadratic |
Formula. Because the discriminant is negative, the equation solution. So, the original system has no (real) solution. x2 x 1 0 has no (real) Now try Exercise 27. 333202_0701.qxd 12/5/05 9:39 AM Page 500 500 Chapter 7 Systems of Equations and Inequalities Te c h n o l o g y Most graphing utilities have builtin features that approximate the point(s) of intersection of two graphs. Typically, you must enter the equations of the graphs and visually locate a point of intersection before using the intersect feature. Use this feature to find the points of intersection of the graphs in Figures 7.1 to 7.3. Be sure to adjust your viewing window so that you see all the points of intersection. Graphical Approach to Finding Solutions From Examples 2, 3, and 4, you can see that a system of two equations in two unknowns can have exactly one solution, more than one solution, or no solution. By using a graphical method, you can gain insight about the number of solutions and the location(s) of the solution(s) of a system of equations by graphing each of the equations in the same coordinate plane. The solutions of the system correspond to the points of intersection of the graphs. For instance, the two equations in Figure 7.1 graph as two lines with a single point of intersection; the two equations in Figure 7.2 graph as a parabola and a line with two points of intersection; and the two equations in Figure 7.3 graph as a line and a parabola that have no points of intersection. y −1 −2 (2, 0) 1 2 x x 2 + 3y = 2 −1 y y = x2 − x − 1 y −x + y = 4 4 (2, 10, − 1) −3 1 −1 −2 x2 + y = 3 x 1 3 One intersection point FIGURE 7.1 Two intersection points FIGURE 7.2 No intersection points FIGURE 7.3 Example 5 Solving a System of Equations Graphically Solve the system of equations. y ln x Equation 1 x y 1 Equation 2 y 1 x + y = 1 y = ln x Solution Sketch the graphs of the two equations. From the graphs of these equations, it is is the solution clear that there is only one point of intersection and that x point (see Figure 7.4). You can confirm this |
by substituting 1 for and 0 for in both equations. 1, 0 y (1, 0) 1 2 x Check (1, 0) in Equation 1: y ln x 0 ln 1 Write Equation 1. Equation 1 checks. ✓ −1 FIGURE 7.4 Check (1, 0) in Equation 2: x y 1 1 0 1 Write Equation 2. Equation 2 checks. ✓ Now try Exercise 33. Example 5 shows the value of a graphical approach to solving systems of equations in two variables. Notice what would happen if you tried only the It substitution method in Example 5. You would obtain the equation would be difficult to solve this equation for using standard algebraic techniques. x ln x 1. x 333202_0701.qxd 12/5/05 9:39 AM Page 501 Section 7.1 Linear and Nonlinear Systems of Equations 501 Applications x C The total cost of producing units of a product typically has two components— the initial cost and the cost per unit. When enough units have been sold so that C, the sales are said to have reached the the total revenue break-even point. You will find that the break-even point corresponds to the point of intersection of the cost and revenue curves. equals the total cost R Example 6 Break-Even Analysis A shoe company invests $300,000 in equipment to produce a new line of athletic footwear. Each pair of shoes costs $5 to produce and is sold for $60. How many pairs of shoes must be sold before the business breaks even? Solution The total cost of producing units is x Total cost Cost per unit Number of units Initial cost C 5x 300,000. Equation 1 The revenue obtained by selling units is x Total revenue Price per unit Number of units R 60x. Equation 2 Because the break-even point occurs when system of equations to solve is R C, you have C 60x, and the C 5x 300,000 C 60x. Now you can solve by substitution. 60x 5x 300,000 55x 300,000 x 5455 Substitute 60x for C in Equation 1. Subtract 5x from each side. Divide each side by 55. Break-Even Analysis 600,000 500,000 Break-even point: 5455 units 400,000 R = 60x Profit 300,000 200,000 100,000 Loss C = 5x + 300,000 ),000 3,000 Number of units 9,000 |
x So, the company must sell about 5455 pairs of shoes to break even. Note in Figure 7.5 that revenue less than the break-even point corresponds to an overall loss, whereas revenue greater than the break-even point corresponds to a profit. FIGURE 7.5 Now try Exercise 63. Another way to view the solution in Example 6 is to consider the profit function P R C. The break-even point occurs when the profit is 0, which is the same as saying that R C. 333202_0701.qxd 12/5/05 9:39 AM Page 502 502 Chapter 7 Systems of Equations and Inequalities Example 7 Movie Ticket Sales The weekly ticket sales for a new comedy movie decreased each week. At the same time, the weekly ticket sales for a new drama movie increased each week. (in millions of dollars) for Models that approximate the weekly ticket sales each movie are S S 60 S 10 8x 4.5x Comedy Drama x represents the number of weeks each movie was in theaters, with x 0 where corresponding to the ticket sales during the opening weekend. After how many weeks will the ticket sales for the two movies be equal? Algebraic Solution Because the second equation has already been solved for in terms of x, solve for S substitute this value into the first equation and x, as follows. 10 4.5x 60 8x 4.5x 8x 60 10 12.5x 50 x 4 Substitute for S in Equation 1. Add 8x and 10 to each side. Combine like terms. Divide each side by 12.5. So, the weekly ticket sales for the two movies will be equal after 4 weeks. Numerical Solution You can create a table of values for each model to determine when the ticket sales for the two movies will be equal. Number of weeks, x Sales, S (comedy) Sales, S (drama) 0 1 2 3 4 5 6 60 52 44 36 28 20 12 10 14.5 19 23.5 28 32.5 37 Now try Exercise 65. So, from the table above, you can see that the weekly ticket sales for the two movies will be equal after 4 weeks. W RITING ABOUT MATHEMATICS Interpreting Points of Intersection You plan to rent a 14-foot truck for a two-day local move. At truck rental agency A, you can rent a truck for $29.95 per day plus $0.49 per mile. At agency |
B, you can rent a truck for $50 per day plus $0.25 per mile. a. Write a total cost equation in terms of and x y for the total cost of renting the truck from each agency. b. Use a graphing utility to graph the two equations in the same viewing window and find the point of intersection. Interpret the meaning of the point of intersection in the context of the problem. c. Which agency should you choose if you plan to travel a total of 100 miles during the two-day move? Why? d. How does the situation change if you plan to drive 200 miles during the two-day move? 333202_0701.qxd 12/5/05 9:39 AM Page 503 Section 7.1 Linear and Nonlinear Systems of Equations 503 7.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. A set of two or more equations in two or more variables is called a ________ of ________. 2. A ________ of a system of equations is an ordered pair that satisfies each equation in the system. 3. Finding the set of all solutions to a system of equations is called ________ the system of equations. 4. The first step in solving a system of equations by the method of ________ is to solve one of the equations for one variable in terms of the other variable. 5. Graphically, the solution of a system of two equations is the ________ of ________ of the graphs of the two equations. 6. In business applications, the point at which the revenue equals costs is called the ________ point. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, determine whether each ordered pair is a solution of the system of equations. 9. 2x y x2 y2 5 25 10. x y 0 x3 5x y 0 y 1. 2. 3. 4. 4x y 1 6x y 6 4x2 y 3 x y 11 y 2e x 3x y 2 log x 3 y 9x y 28 1 9 (a) (c) (a) (c) (a) (c) (a |
) (c) 0, 3 3 2, 2 2, 13 3 2, 31 3 2, 0 0, 3 9, 37 9 1, 3 (b) (d) (b) (d) 1, 4 1 2, 3 2, 9 7 4, 37 4 0, 2 1, 2 10, 2 (b) (d) 2, 4 (d) (b) In Exercises 5–14, solve the system by the method of substitution. Check your solution graphically. 5. 2x y 6 x y 0 y 6. x y 4 x 2y 5 y 6 4 2 −2 −2 2 4 6 7. x y 4 x2 y 2 y 6 4 −2 2 4 x x 6 4 2 −2 2 8. 3x y 2 x3 2 y 0 y 8 6 −2 −2 −4 2 x x y 8 6 2 −6 −2 −6 x 6 8 −4 2 −2 −4 x 4 11. x2 y 0 x2 4x y 0 12. y 2x2 2 y 2x 4 2x2 1 y 2 −2 2 x −4 y 1 x 1 13. y x3 3x2 1 y x2 3x 1 14. y x3 3x2 4 y 2x 4 y 1 − 333202_0701.qxd 12/5/05 9:39 AM Page 504 504 Chapter 7 Systems of Equations and Inequalities In Exercises 15–28, solve the system by the method of substitution. In Exercises 49–60, solve the system graphically or algebraically. Explain your choice of method. 15. 17. 19. 21. 23. 25. 27. x y 0 5x 3y 10 2x y 2 0 4x y 5 0 1.5x 0.8y 2.3 0.3x 0.2y 0.1 1 1 8 2 y 20 y 6x 3 5y 5 x 7 6 y x2 y 0 2x y 0 x y 1 x2 y 4 5x x 16. 18. 20. 22. 24. 26. 28. x 2y 1 5x 4y 23 6x 3y 4 0 x 2y 4 0 0.5x 3.2y 9.0 0.2x 1.6y 3.6 1 2 x 3 4 y 10 4 x y 4 2 3x y 2 2x 3y 6 |
x 2y 0 3x y2 0 y x y x3 3x2 2x 3 In Exercises 29– 42, solve the system graphically. 29. 31. 33. 34. 35. 37. 39. 41. x 2y 2 3x y 15 x 3y 2 5x 3y 17 x y 4 x2 y2 4x 0 x y 3 x2 6x 27 y2 0 x y 3 0 x2 4x 7 y 7x 8y 24 x 8y 8 3x 2y 0 x2 y2 4 x2 25 0 y2 16y 3x2 30. 32. x y 0 3x 2y 10 x 2y 1 x y 2 36. 38. 40. 42. 0 1 2 1 y2 4x 11 2 x y x y 0 5x 2y 6 2x y 3 0 x2 y2 4x 0 x2 y2 25 x 82 y2 41 In Exercises 43– 48, use a graphing utility to solve the system of equations. Find the solution accurate to two decimal places. 49. 51. 53. 55. 57. 59. y 2x y x2 1 3x 7y 6 0 x2 y2 4 x 2y 4 x2 y 0 y ex 1 y ln x 3 y x 4 2x2 1 y 1 x2 xy 1 0 2x 4y 7 0 50. 52. 54. 56. 58. 60. x y 4 x2 y 2 x2 y2 25 2x y 10 y x 13 y x 1 x2 y 4 ex y 0 y x 3 2x2 x 1 y x2 3x 1 x 2y 1 y x 1 Break-Even Analysis necessary to break even producing x R C units and the revenue In Exercises 61 and 62, find the sales for the cost of obtained by selling units. (Round to the nearest whole unit.) C R x 61. 62. C 8650x 250,000, C 5.5x 10,000, R 9950x R 3.29x 63. Break-Even Analysis A small software company invests $16,000 to produce a software package that will sell for $55.95. Each unit can be produced for $35.45. (a) How many units must be sold to break even? (b) How many units must be sold to make a profit of $60,000? 64. Break-Even Analysis A |
small fast-food restaurant invests $5000 to produce a new food item that will sell for $3.49. Each item can be produced for $2.16. (a) How many items must be sold to break even? (b) How many items must be sold to make a profit of $8500? 65. DVD Rentals The weekly rentals for a newly released DVD of an animated film at a local video store decreased each week. At the same time, the weekly rentals for a newly released DVD of a horror film increased each week. Models that approximate the weekly rentals for each DVD are R 43. 45. 46. 47. y e x x y 1 0 x 2y 8 y log2 x y 2 lnx 1 3y 2x 9 x2 y2 169 x2 8y 104 48. 44. y 4ex y 3x 8 0 R 360 24x R 24 18x Animated film Horror film x where the store, with represents the number of weeks each DVD was in x 1 corresponding to the first week. x2 y2 4 2x2 y 2 (a) After how many weeks will the rentals for the two movies be equal? (b) Use a table to solve the system of equations numeri- cally. Compare your result with that of part (a). 333202_0701.qxd 12/5/05 9:39 AM Page 505 Section 7.1 Linear and Nonlinear Systems of Equations 505 66. CD Sales The total weekly sales for a newly released rock CD increased each week. At the same time, the total weekly sales for a newly released rap CD decreased each S week. Models that approximate the total weekly sales (in thousands of units) for each CD are 70. Log Volume You are offered two different rules for estimating the number of board feet in a 16-foot log. (A board foot is a unit of measure for lumber equal to a board 1 foot square and 1 inch thick.) The first rule is the Doyle Log Rule and is modeled by S 25x 100 S 50x 475 Rock CD Rap CD x represents the number of weeks each CD was in corresponding to the CD sales on the where stores, with day each CD was first released in stores. x 0 (a) After how many weeks will the sales for the two CDs D 42, V1 5 ≤ D ≤ 40 and the other is the Scribner Log Rule and is modeled by V2 0.79D 2 2D 4, 5 ≤ D ≤ |
40 D where volume (in board feet). is the diameter (in inches) of the log and V is its be equal? (a) Use a graphing utility to graph the two log rules in the (b) Use a table to solve the system of equations numeri- same viewing window. cally. Compare your result with that of part (a). (b) For what diameter do the two scales agree? 67. Choice of Two Jobs You are offered two jobs selling dental supplies. One company offers a straight commission of 6% of sales. The other company offers a salary of $350 per week plus 3% of sales. How much would you have to sell in a week in order to make the straight commission offer better? 68. Supply and Demand The supply and demand curves for a business dealing with wheat are Supply: p 1.45 0.00014x 2 Demand: p 2.388 0.007x 2 (c) You are selling large logs by the board foot. Which scale would you use? Explain your reasoning. Model It 71. Data Analysis: Renewable Energy The table shows (in trillions of Btus) of solar energy the consumption and wind energy in the United States from 1998 to (Source: Energy Information Administration) 2003. C p where is the is the price in dollars per bushel and quantity in bushels per day. Use a graphing utility to graph the supply and demand equations and find the market equilibrium. (The market equilibrium is the point of intersection of the graphs for x > 0. ) x 69. Investment Portfolio A total of $25,000 is invested in two funds paying 6% and 8.5% simple interest. (The 6% investment has a lower risk.) The investor wants a yearly interest income of $2000 from the two investments. (a) Write a system of equations in which one equation represents the total amount invested and the other equation represents the $2000 required in interest. Let x y and represent the amounts invested at 6% and 8.5%, respectively. (b) Use a graphing utility to graph the two equations in the same viewing window. As the amount invested at 6% increases, how does the amount invested at 8.5% change? How does the amount of interest income change? Explain. (c) What amount should be invested at 6% to meet the requirement of $2000 per year in interest? Year Solar, C Wind, C 1998 1999 2000 2001 2002 2003 70 69 66 65 64 63 31 |
46 57 68 105 108 (a) Use the regression feature of a graphing utility to find a quadratic model for the solar energy consumption data and a linear model for the wind represent the year, energy consumption data. Let with corresponding to 1998. t 8 t (b) Use a graphing utility to graph the data and the two models in the same viewing window. (c) Use the graph from part (b) to approximate the point of intersection of the graphs of the models. Interpret your answer in the context of the problem. (d) Approximate the point of intersection of the graphs of the models algebraically. (e) Compare your results from parts (c) and (d). (f) Use your school’s library, the Internet, or some other reference source to research the advantages and disadvantages of using renewable energy. 333202_0701.qxd 12/5/05 9:39 AM Page 506 506 Chapter 7 Systems of Equations and Inequalities 81. Writing List and explain the steps used to solve a system of equations by the method of substitution. 82. Think About It When solving a system of equations by substitution, how do you recognize that the system has no solution? 83. Exploration Find an equation of a line whose graph intersects the graph of the parabola at (a) two points, (b) one point, and (c) no points. (There is more than one correct answer.) y x 2 84. Conjecture Consider the system of equations y b x y x b. (a) Use a graphing utility to graph the system for b 1, 2, 3, and 4. (b) For a fixed even value of make a conjecture about the number of points of intersection of the graphs in part (a). b > 1, Skills Review In Exercises 85–90, find the general form of the equation of the line passing through the two points. 85. 86. 87. 88. 89. 90. 2, 7, 5, 5 3.5, 4, 10, 6 6, 3, 10, 3 4, 2, 4, 5 5, 0, 4, 6 3 7 3, 8, 5 2, 1 2 In Exercises 91–94, find the domain of the function and identify any horizontal or vertical asymptotes. 91. 92. 93. 94. f x 5 x 6 f x 2x 7 3x 2 f x x2 |
2 x2 16 f x 3 2 x2 72. Data Analysis: Population The table shows the popula(in thousands) of Alabama and Colorado from 1999 P tions to 2003. (Source: U.S. Census Bureau) Year Alabama, P Colorado, P 1999 2000 2001 2002 2003 4430 4447 4466 4479 4501 4226 4302 4429 4501 4551 (a) Use the regression feature of a graphing utility to find linear models for each set of data. Graph the models in the same viewing window. Let represent the year, with t corresponding to 1999. t 9 (b) Use your graph from part (a) to approximate when the population of Colorado exceeded the population of Alabama. (c) Verify your answer from part (b) algebraically. Geometry rectangle meeting the specified conditions. In Exercises 73–76, find the dimensions of the 73. The perimeter is 30 meters and the length is 3 meters greater than the width. 74. The perimeter is 280 centimeters and the width is 20 centimeters less than the length. 75. The perimeter is 42 inches and the width is three-fourths the length. 76. The perimeter is 210 feet and the length is 11 2 times the width. 77. Geometry What are the dimensions of a rectangular tract of land if its perimeter is 40 kilometers and its area is 96 square kilometers? 78. Geometry What are the dimensions of an isosceles right triangle with a two-inch hypotenuse and an area of 1 square inch? Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 79 and 80, determine whether 79. In order to solve a system of equations by substitution, you in one of the two equations and y must always solve for then back-substitute. 80. If a system consists of a parabola and a circle, then the system can have at most two solutions. 333202_0702.qxd 12/5/05 9:41 AM Page 507 7.2 Two-Variable Linear Systems Section 7.2 Two-Variable Linear Systems 507 What you should learn • Use the method of elimination to solve systems of linear equations in two variables. • Interpret graphically the numbers of solutions of systems of linear equations in two variables. • Use systems of linear equations in two variables to model and solve real-life problems. Why you should learn it You can use systems of equations in two variables to model and |
solve real-life problems. For instance, in Exercise 63 on page 517, you will solve a system of equations to find a linear model that represents the relationship between wheat yield and amount of fertilizer applied. So, © Bill Stormont /Corbis The Method of Elimination In Section 7.1, you studied two methods for solving a system of equations: substitution and graphing. Now you will study the method of elimination. The key step in this method is to obtain, for one of the variables, coefficients that differ only in sign so that adding the equations eliminates the variable. 3x 5y 7 3x 2y 1 3y 6 Equation 1 Equation 2 Add equations. Note that by adding the two equations, you eliminate the -terms and obtain a which you can single equation in Solving this equation for produces then back-substitute into one of the original equations to solve for x y 2, y. x. y Example 1 Solving a System of Equations by Elimination Solve the system of linear equations. 3x 2y 4 5x 2y 8 Equation 1 Equation 2 y Solution Because the coefficients of differ only in sign, you can eliminate the -terms by adding the two equations. 3x 2y 4 5x 2y 8 12 Write Equation 1. Write Equation 2. Add equations. y By back-substituting this value into Equation 1, you can solve for y. Write Equation 1. Substitute 3 2 for x. Simplify. Solve for y.. Check this in the original system, as follows. 8x x 3 2. 3x 2y 4 2y 4 33 9 2y 4 2 2 y 1 4 3 2, 1 4 The solution is Check 33 2 21 4 1 9 2 2 21 4 15 1 2 2? 4 4? 8 8 Exploration Use the method of substitution to solve the system in Example 1. Which method is easier? 53 2 Substitute into Equation 1. Equation 1 checks. ✓ Substitute into Equation 2. Equation 2 checks. ✓ Now try Exercise 11. 333202_0702.qxd 12/5/05 9:41 AM Page 508 508 Chapter 7 Systems of Equations and Inequalities Method of Elimination To use the method of elimination to solve a system of two linear equations x in and perform the following steps. y, 1. Obtain coefficients for y terms of one or both equations by suitably chosen |
constants. (or ) that differ only in sign by multiplying all x 2. Add the equations to eliminate one variable, and solve the resulting equation. 3. Back-substitute the value obtained in Step 2 into either of the original equations and solve for the other variable. 4. Check your solution in both of the original equations. Example 2 Solving a System of Equations by Elimination Solve the system of linear equations. 2x 3y 7 3x y 5 Equation 1 Equation 2 Solution For this system, you can obtain coefficients that differ only in sign by multiplying Equation 2 by 3. 2x 3y 7 3x y 5 So, you can see that y. you can solve for 2x 3y 7 9x 3y 15 22 11x Write Equation 1. Multiply Equation 2 by 3. Add equations. x 2. By back-substituting this value of x into Equation 1, 2x 3y 7 22 3y 7 3y 3 y 1 2, 1. The solution is Check Write Equation 1. Substitute 2 for x. Combine like terms. Solve for y. Check this in the original system, as follows. 2x 3y 7 22 31? 7 4 3 7 3x y 5 32 1? 5 6 1 5 Now try Exercise 13. Write original Equation 1. Substitute into Equation 1. Equation 1 checks. ✓ Write original Equation 2. Substitute into Equation 2. Equation 2 checks. ✓ Exploration Rewrite each system of equations in slope-intercept form and sketch the graph of each system. What is the relationship between the slopes of the two lines and the number of points of intersection? a. b. 5x y 1 x y 5 4x 3y 1 8x 6y 2 c. x 2y 3 x 2y 8 333202_0702.qxd 12/5/05 9:41 AM Page 509 Section 7.2 Two-Variable Linear Systems 509 In Example 2, the two systems of linear equations (the original system and the system obtained by multiplying by constants) 2x 3y 7 3x y 5 and 2x 3y 7 9x 3y 15 are called equivalent systems because they have precisely the same solution set. The operations that can be performed on a system of linear equations to produce an equivalent system are (1) interchanging any two equations, (2) multiplying an equation by a nonzero constant, and |
(3) adding a multiple of one equation to any other equation in the system. Example 3 Solving the System of Equations by Elimination Solve the system of linear equations. 5x 3y 9 2x 4y 14 Equation 1 Equation 2 Algebraic Solution You can obtain coefficients that differ only in sign by multiplying Equation 1 by 4 and multiplying Equation 2 by 3. 5x 3y 9 2x 4y 14 20x 12y 36 6x 12y 42 78 26x Multiply Equation 1 by 4. Multiply Equation 2 by 3. Add equations. From this equation, you can see that of into Equation 2, you can solve for x x 3. y. 2x 4y 14 23 4y 14 4y 8 y 2 By back-substituting this value Write Equation 2. Substitute 3 for x. Combine like terms. Solve for y. The solution is 3, 2. Check this in the original system. Now try Exercise 15. 2 y2 1 y. y1 Graphical Solution Then use a Solve each equation for 5 3x 3 graphing utility to graph 2x 7 in the same viewing and window. Use the intersect feature or the zoom and trace features to approximate the point of intersection of the graphs. From the graph in Figure 7.6, you can see that the point of intersection is You can determine that this is the exact solution by checking in both equations. 3, 2. 3, 2 5 y1 = − x + 3 3 3 7 1 y2 = x − 2 7 2 −5 −5 FIGURE 7.6 You can check the solution from Example 3 as follows. 53 32? 15 23 42? 9 6 9 14 8 14 6 Substitute 3 for x and Equation 1 checks. ✓ 2 Substitute 3 for x and Equation 2 checks. ✓ 2 for y in Equation 1. for y in Equation 2. Keep in mind that the terminology and methods discussed in this section apply only to systems of linear equations. 333202_0702.qxd 12/5/05 9:41 AM Page 510 510 Chapter 7 Systems of Equations and Inequalities Graphical Interpretation of Solutions It is possible for a general system of equations to have exactly one solution, two or more solutions, or no solution. If a system of linear equations has two different solutions, it must have an infinite number of solutions. Graphical Interpretations of Solutions For a |
system of two linear equations in two variables, the number of solutions is one of the following. Number of Solutions Graphical Interpretation Slopes of Lines 1. Exactly one solution The two lines intersect at one point. The slopes of the two lines are not equal. 2. Infinitely many solutions The two lines coincide (are identical). The slopes of the two lines are equal. 3. No solution The two lines are parallel. The slopes of the two lines are equal. A system of linear equations is consistent if it has at least one solution. A consistent system with exactly one solution is independent, whereas a consistent system with infinitely many solutions is dependent. A system is inconsistent if it has no solution. Example 4 Recognizing Graphs of Linear Systems Match each system of linear equations with its graph in Figure 7.7. Describe the number of solutions and state whether the system is consistent or inconsistent. 2x 3y 3 4x 6y 6 a. i. b. ii. y 4 2 −2 2 4 x −2 −4 FIGURE 7.7 2x 3y 3 x 2y 5 c. 2x 3y 3 4x 6y 6 iii. y 4 2 x 2 4 −2 2 4 x −2 −4 y 4 2 −2 −4 A comparison of the slopes of two lines gives useful information about the number of solutions of the corresponding system of equations. To solve a system of equations graphically, it helps to begin by writing the equations in slope-intercept form. Try doing this for the systems in Example 4. Solution a. The graph of system (a) is a pair of parallel lines (ii). The lines have no point of intersection, so the system has no solution. The system is inconsistent. b. The graph of system (b) is a pair of intersecting lines (iii). The lines have one point of intersection, so the system has exactly one solution. The system is consistent. c. The graph of system (c) is a pair of lines that coincide (i). The lines have infinitely many points of intersection, so the system has infinitely many solutions. The system is consistent. Now try Exercises 31–34. 333202_0702.qxd 12/5/05 9:41 AM Page 511 Section 7.2 Two-Variable Linear Systems 511 In Examples 5 and 6, note how you can use the method of elimination to determine that a system of linear equations has no solution or infinitely many solutions. Example 5 |
No-Solution Case: Method of Elimination Solve the system of linear equations. x 2y 3 2x 4y 1 Equation 1 Equation 2 Solution To obtain coefficients that differ only in sign, multiply Equation 1 by 2. −2x + 4y = 1 1 3 x x − 2y = 3 x 2y 3 2x 4y 1 2x 4y 6 2x 4y 1 0 7 Multiply Equation 1 by 2. Write Equation 2. False statement Because there are no values of and you can conclude that the system is inconsistent and has no solution. The lines corresponding to the two equations in this system are shown in Figure 7.8. Note that the two lines are parallel and therefore have no point of intersection. for which y x 0 7, y 2 1 −1 −2 FIGURE 7.8 Now try Exercise 19. 0 7, In Example 5, note that the occurrence of a false statement, such as indicates that the system has no solution. In the next example, note that the occurrence of a statement that is true for all values of the variables, such as 0 0, indicates that the system has infinitely many solutions. Example 6 Many-Solution Case: Method of Elimination Solve the system of linear equations. 2x y 1 4x 2y 2 Equation 1 Equation 2 Solution To obtain coefficients that differ only in sign, multiply Equation 2 by 1 2. 2x y 1 4x 2y 2 2x y 1 2x y 1 0 0 Write Equation 1. Multiply Equation 2 by 1 2. Add equations. Because the two equations turn out to be equivalent (have the same solution set), you can conclude that the system has infinitely many solutions. The solution set x, y as shown in Figure 7.9. consists of all points x a, a where Letting is any real number, you can see that the solutions to the a, 2a 1. system are lying on the line 2x y 1, Now try Exercise 23. y 3 2 1 (2, 3) 2x − y = 1 (1, 1) −1 1 2 3 x −1 FIGURE 7.9 333202_0702.qxd 12/5/05 9:41 AM Page 512 512 Chapter 7 Systems of Equations and Inequalities Te c h n o l o g y The general solution of the linear system ax by c dx ey f and x ce bf ae b |
d is y af cdae bd. If ae bd 0, the system does not have a unique solution. A graphing utility program (called Systems of Linear Equations) for solving such a system can be found at our website college.hmco.com. Try using the program for your graphing utility to solve the system in Example 7. Example 7 illustrates a strategy for solving a system of linear equations that has decimal coefficients. Example 7 A Linear System Having Decimal Coefficients Solve the system of linear equations. 0.02x 0.05y 0.38 0.03x 0.04y 1.04 Equation 1 Equation 2 Solution Because the coefficients in this system have two decimal places, you can begin by multiplying each equation by 100. This produces a system in which the coefficients are all integers. 2x 5y 38 3x 4y 104 Revised Equation 1 Revised Equation 2 Now, to obtain coefficients that differ only in sign, multiply Equation 1 by 3 and multiply Equation 2 by 2x 5y 38 3x 4y 104 Multiply Equation 1 by 3. Multiply Equation 2 by 2. 2. 6x 15y 114 6x 8y 208 23y 322 Add equations. So, you can conclude that 322 23 y 14. Back-substituting this value into revised Equation 2 produces the following. 3x 4y 104 3x 414 104 3x 48 x 16 16, 14. The solution is Check Write revised Equation 2. Substitute 14 for y. Combine like terms. Solve for x. Check this in the original system, as follows. 0.02x 0.05y 0.38 0.0216 0.0514? 0.38 0.32 0.70 0.38 0.03x 0.04y 1.04 0.0316 0.0414? 1.04 0.48 0.56 1.04 Now try Exercise 25. Write original Equation 1. Substitute into Equation 1. Equation 1 checks. ✓ Write original Equation 2. Substitute into Equation 2. Equation 2 checks. ✓ 333202_0702.qxd 12/5/05 9:41 AM Page 513 Section 7.2 Two-Variable Linear Systems 513 Applications At this point, you may be asking the question “How can I tell which application problems can be solved using a system of linear equations?” The answer comes |
from the following considerations. 1. Does the problem involve more than one unknown quantity? 2. Are there two (or more) equations or conditions to be satisfied? If one or both of these situations occur, the appropriate mathematical model for the problem may be a system of linear equations. Example 8 An Application of a Linear System An airplane flying into a headwind travels the 2000-mile flying distance between Chicopee, Massachusetts and Salt Lake City, Utah in 4 hours and 24 minutes. On the return flight, the same distance is traveled in 4 hours. Find the airspeed of the plane and the speed of the wind, assuming that both remain constant. Solution The two unknown quantities are the speeds of the wind and the plane. If speed of the plane and is the speed of the wind, then r2 r1 is the Original flight WIND r1 − r2 Return flight WIND r1 + r2 FIGURE 7.10 r1 r1 r2 r2 speed of the plane against the wind speed of the plane with the wind as shown in Figure 7.10. Using the formula speeds, you obtain the following equations. distance ratetime for these two 2000 r1 r2 4 24 60 2000 r1 r2 4 These two equations simplify as follows. 5000 11r1 500 r1 11r2 r2 Equation 1 Equation 2 To solve this system by elimination, multiply Equation 2 by 11. 5000 11r1 500 r1 11r2 11r2 11r2 r2 Write Equation 1. Multiply Equation 2 by 11. 5000 11r1 5500 11r1 10,500 22r1 Add equations. So, r1 r2 10,500 5250 11 22 500 5250 11 250 11 477.27 miles per hour Speed of plane 22.73 miles per hour. Speed of wind Check this solution in the original statement of the problem. Now try Exercise 43. 333202_0702.qxd 12/5/05 9:41 AM Page 514 514 Chapter 7 Systems of Equations and Inequalities In a free market, the demands for many products are related to the prices of the products. As the prices decrease, the demands by consumers increase and the amounts that producers are able or willing to supply decrease. Example 9 Finding the Equilibrium Point The demand and supply functions for a new type of personal digital assistant are p 150 0.00001x p 60 0.00002x Demand equation Supply equation Equilibrium p |
(3,000,000, 120) Demand p is the price in dollars and x where equilibrium point for this market. The equilibrium point is the price number of units represents the number of units. Find the and that satisfy both the demand and supply equations. p x Solution p Because is written in terms of the supply equation into the demand equation. x, begin by substituting the value of given in p Supply p 150 0.00001x 60 0.00002x 150 0.00001x Write demand equation. Substitute 60 0.00002x for p. 0.00003x 90 Combine like terms. x 3,000,000 Solve for x. 150 125 100 75 50 25 ),000,000 3,000,000 Number of units x FIGURE 7.11 So, the equilibrium point occurs when the demand and supply are each 3 million units. (See Figure 7.11.) The price that corresponds to this -value is obtained by back-substituting into either of the original equations. For instance, back-substituting into the demand equation produces x 3,000,000 x p 150 0.000013,000,000 150 30 $120. The solution is 3,000,000, 120. You can check this as follows. into the demand equation. Check Substitute 3,000,000, 120 p 150 0.00001x 150 0.000013,000,000 120? 120 120 Substitute 3,000,000, 120 p 60 0.00002x into the supply equation. 60 0.000023,000,000 120? 120 120 Now try Exercise 45. Write demand equation. Substitute 120 for p and 3,000,000 for x. Solution checks in demand equation. ✓ Write supply equation. Substitute 120 for p and 3,000,000 for x. Solution checks in supply equation. ✓ 333202_0702.qxd 12/5/05 9:41 AM Page 515 Section 7.2 Two-Variable Linear Systems 515 7.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The first step in solving a system of equations by the method of ________ is to obtain coefficients for x (or ) that differ only in sign. y 2. Two systems of equations that have the same solution set are called ________ systems. 3. A system of linear equations that has at least one solution is called ________, whereas a |
system of linear equations that has no solution is called ________. 4. In business applications, the ________ ________ is defined as the price and the number of units p x that satisfy both the demand and supply equations. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–10, solve the system by the method of elimination. Label each line with its equation. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 7. 3x 2y 5 6x 4y 10 8. 9x 3y 15 3x y 5 1. 2x y 5 x y 1 y 4 2 −2 2 4 6 x −4 3. x y 0 3x 2y 1 y 4 x 2 4 −4 −2 −2 −4 5. x y 2 2x 2y 5 y 4 x 2 4 −2 −2 2. x 3y 1 x 2y 2 −2 −4 −2 −2 x 2 9. 9x 3y 1 3x 6y 5 y 4. 2x y 3 4x 3y 21 y 6 4 −4 x 2 10. 5x 3y 18 2x 6y 1 y 4 2 −2 x 2 x 2 4 −6 −4 −2 −2 −4 2 4 x In Exercises 11–30, solve the system by the method of elimination and check any solutions algebraically. 6. 3x 2y 3 6x 4y 14 y x 4 −2 −2 −4 11. 13. 15. 17. 19. x 2y 4 x 2y 1 2x 3y 18 5x y 11 3x 2y 10 2x 5y 3 5u 6v 24 3u 5v 18 9 6 5x 5y 4 6y 3 9x 12. 14. 16. 18. 3x 5y 2 2x 5y 13 x 7y 12 3x 5y 10 2r 4s 5 16r 50s 55 3x 11y 4 2x 5y 9 y 1 4x 8 3y 3 4x 8 9 20. 3 y 6 4 2 −2 333202_0702.qxd 12/5/05 9:41 AM Page 516 516 Chapter 7 Systems of Equations and Inequalities 21. 23. 25. 27. 29. 4 x y 6 y 3 12 1 |
3 x 5x 6y 24y 20x 0.05x 0.03y 0.21 0.07x 0.02y 0.16 4b 3m 3 3b 11m 13 x 3 y 1 3 2x y 12 1 4 22. 24. 26. 28. 30 8y 7x 6 12 16y 14x 0.2x 0.5y 27.8 0.3x 0.4y 68.7 2x 5y 8 5x 8y 10 x 1 y 2 3 x 2y 5 4 2 In Exercises 31–34, match the system of linear equations with its graph. Describe the number of solutions and state whether the system is consistent or inconsistent. [The graphs are labeled (a), (b), (c) and (d).] (b) y 4 2 (d) (a) y 4 2 −2 2 4 −4 (c) y −6 2 −2 −4 2 4 31. 33. 2x 5y 0 x y 3 2x 5y 0 2x 3y 4 32. 34. 7x 6y 4 14x 12y 8 7x 6y 6 7x 6y 4 In Exercises 35–42, use any method to solve the system. 35. 37. 39. 3x 5y 7 2x y 9 y 2x 5 y 5x 11 x 5y 21 6x 5y 21 36. 38. 40. x 3y 17 4x 3y 7 7x 3y 16 y x 2 y 3x 8 y 15 2x 41. 2x 8y 19 y x 3 42. 4x 3y 6 5x 7y 1 43. Airplane Speed An airplane flying into a headwind travels the 1800-mile flying distance between Pittsburgh, Pennsylvania and Phoenix, Arizona in 3 hours and 36 minutes. On the return flight, the distance is traveled in 3 hours. Find the airspeed of the plane and the speed of the wind, assuming that both remain constant. 44. Airplane Speed Two planes start from Los Angeles International Airport and fly in opposite directions. The second plane starts hour after the first plane, but its speed is 80 kilometers per hour faster. Find the airspeed of each plane if 2 hours after the first plane departs the planes are 3200 kilometers apart. 1 2 In Exercises 45– 48, Supply and Demand the equilibrium point of the demand and supply equations. The equilibrium point is the price p and number |
of units x that satisfy both the demand and supply equations. find Demand p 50 0.5x p 100 0.05x p 140 0.00002x p 400 0.0002x 45. 46. 47. 48. Supply p 0.125x p 25 0.1x p 80 0.00001x p 225 0.0005x 49. Nutrition Two cheeseburgers and one small order of French fries from a fast-food restaurant contain a total of 850 calories. Three cheeseburgers and two small orders of French fries contain a total of 1390 calories. Find the caloric content of each item. 50. Nutrition One eight-ounce glass of apple juice and one eight-ounce glass of orange juice contain a total of 185 milligrams of vitamin C. Two eight-ounce glasses of apple juice and three eight-ounce glasses of orange juice contain a total of 452 milligrams of vitamin C. How much vitamin C is in an eight-ounce glass of each type of juice? 51. Acid Mixture Ten liters of a 30% acid solution is obtained by mixing a 20% solution with a 50% solution. (a) Write a system of equations in which one equation represents the amount of final mixture required and the other represents the percent of acid in the final mixture. x y Let and represent the amounts of the 20% and 50% solutions, respectively. (b) Use a graphing utility to graph the two equations in part (a) in the same viewing window. As the amount of the 20% solution increases, how does the amount of the 50% solution change? (c) How much of each solution is required to obtain the specified concentration of the final mixture? 333202_0702.qxd 12/5/05 9:41 AM Page 517 52. Fuel Mixture Five hundred gallons of 89 octane gasoline is obtained by mixing 87 octane gasoline with 92 octane gasoline. (a) Write a system of equations in which one equation represents the amount of final mixture required and the other represents the amounts of 87 and 92 octane gasolines in the final mixture. Let and represent the numbers of gallons of 87 octane and 92 octane gasolines, respectively. x y (b) Use a graphing utility to graph the two equations in part (a) in the same viewing window. As the amount of 87 octane gasoline increases, how does the amount of 92 octane gasoline change? (c) How much of |
each type of gasoline is required to obtain the 500 gallons of 89 octane gasoline? 53. Investment Portfolio A total of $12,000 is invested in two corporate bonds that pay 7.5% and 9% simple interest. The investor wants an annual interest income of $990 from the investments. What amount should be invested in the 7.5% bond? 54. Investment Portfolio A total of $32,000 is invested in two municipal bonds that pay 5.75% and 6.25% simple interest. The investor wants an annual interest income of $1900 from the investments. What amount should be invested in the 5.75% bond? 55. Ticket Sales At a local high school city championship basketball game, 1435 tickets were sold. A student admission ticket cost $1.50 and an adult admission ticket cost $5.00. The sum of all the total ticket receipts for the basketball game were $3552.50. How many of each type of ticket were sold? 56. Consumer Awareness A department store held a sale to sell all of the 214 winter jackets that remained after the season ended. Until noon, each jacket in the store was priced at $31.95. At noon, the jackets was further reduced to $18.95. After the last jacket was sold, total receipts for the clearance sale were $5108.30. How many jackets were sold before noon and how many were sold after noon? the price of Fitting a Line to Data squares regression line In Exercises 57–62, find the least y ax b for the points x1, y1, x2, y2,..., xn, yn by solving the system for a and b. nb n i1 xia n i1 yi n i1 xib n i1 i a n x2 i1 xi yi Section 7.2 Two-Variable Linear Systems 517 Then use a graphing utility to confirm the result. (If you are unfamiliar with summation notation, look at the discussion in Section 9.1 or in Appendix B at the website for this text at college.hmco.com.) 57. 5b 10a 20.2 10b 30a 50.1 58. 5b 10a 11.7 10b 30a 25.6 y 6 5 4 3 2 1 (4, 5.8) (3, 5.2) (2, 4.2) (1, 2.9) ( |
0, 2.1) −1 1 2 3 4 5 59. 7b 21a 35.1 21b 91a 114.2 y 8 6 2 (5, 5.6) (6, 6) (3, 5) (4, 5.4) (2, 4.6) (1, 4.4) (0, 4.11 −2 (4, 2.8) (2, 2.4) (3, 2.5) (1, 2.1) 2 3 4 5 (0, 1.9) x 60. 6b 15a 23.6 15b 55a 48.8 y 8 (0, 5.4) (1, 4.8) 4 2 (3, 3.5) (5, 2.5) (2, 4.3) (4, 3.1) 2 4 6 x 61. 62. 0, 4, 1, 3, 1, 1, 2, 0 1, 0, 2, 0, 3, 0, 3, 1, 4, 1, 4, 2, 5, 2, 6, 2 63. Data Analysis A farmer used four test plots to determine the relationship between wheat yield (in bushels per acre) x (in hundreds of pounds per and the amount of fertilizer acre). The results are shown in the table. y Fertilizer, x Yield, y 1.0 1.5 2.0 2.5 32 41 48 53 (a) Use the technique demonstrated in Exercises 57–62 to set up a system of equations for the data and to find the least squares regression line y ax b. (b) Use the linear model to predict the yield for a fertilizer application of 160 pounds per acre. 333202_0702.qxd 12/5/05 9:41 AM Page 518 518 Chapter 7 Systems of Equations and Inequalities Model It 64. Data Analysis The table shows the average room for a hotel room in the United States for the (Source: American Hotel rates years 1995 through 2001. & Motel Association) y Year Average room rate, y 1995 1996 1997 1998 1999 2000 2001 $66.65 $70.93 $75.31 $78.62 $81.33 $85.89 $88.27 (a) Use the technique demonstrated in Exercises 57–62 to set up a system of equations for the data and to find the least squares regression line y |
at b. t 5 corresponding to 1995. represent the year, with Let t (b) Use the regression feature of a graphing utility to find a linear model for the data. How does this model compare with the model obtained in part (a)? (c) Use the linear model to create a table of estimated y. values of Compare the estimated values with the actual data. (d) Use the linear model to predict the average room rate in 2002. The actual average room rate in 2002 was $83.54. How does this value compare with your prediction? (e) Use the linear model to predict when the average room rate will be $100.00. Using your result from part (d), do you think this prediction is accurate? Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 65 and 66, determine whether 65. If two lines do not have exactly one point of intersection, then they must be parallel. 66. Solving a system of equations graphically will always give an exact solution. 67. Writing Briefly explain whether or not it is possible for a consistent system of linear equations to have exactly two solutions. 68. Think About It Give examples of a system of linear equations that has (a) no solution and (b) an infinite number of solutions. Think About It In Exercises 69 and 70, the graphs of the two equations appear to be parallel. Yet, when the system is solved algebraically, you find that the system does have a solution. Find the solution and explain why it does not appear on the portion of the graph that is shown. 69. 100y x 200 99y x 198 y 70. 21x 20y 0 13x 12y 120 y 4 −4 −2 2 4 x −4 10 −10 −10 x 10 In Exercises 71 and 72, find the value of k such that the system of linear equations is inconsistent. 71. 4x 8y 3 2x ky 16 72. 15x 3y 6 10x ky 9 Skills Review In Exercises 73–80, solve the inequality and graph the solution on the real number line. 73. 75. 77. 79. 11 6x ≥ 33 8x 15 ≤ 42x 1 x 8 < 10 2x2 3x 35 < 0 74. 76. 78. 80. 2x 3 > 5x 1 6 ≤ 3x 10 < 6 x 10 ≥ 3 3x2 12x > 0 In |
Exercises 81–84, write the expression as the logarithm of a single quantity. 81. 83. ln x ln 6 log9 12 log9 x 82. 84. ln x 5 lnx 3 1 4 log6 3x In Exercises 85 and 86, solve the system by the method of substitution. 85. 2x y 4 4x 2y 12 86. 30x 40y 33 0 10x 20y 21 0 87. Make a Decision To work an extended application analyzing the average undergraduate tuition, room, and board charges at private colleges in the United States from 1985 to 2003, visit this text’s website at college.hmco.com. (Data Source: U.S. Dept. of Education) 333202_0703.qxd 12/5/05 9:42 AM Page 519 Section 7.3 Multivariable Linear Systems 519 7.3 Multivariable Linear Systems What you should learn • Use back-substitution to solve linear systems in row-echelon form. • Use Gaussian elimination to solve systems of linear equations. • Solve nonsquare systems of linear equations. • Use systems of linear equations in three or more variables to model and solve real-life problems. Why you should learn it Systems of linear equations in three or more variables can be used to model and solve real-life problems. For instance, in Exercise 71 on page 531, a system of linear equations can be used to analyze the reproduction rates of deer in a wildlife preserve. Row-Echelon Form and Back-Substitution The method of elimination can be applied to a system of linear equations in more than two variables. In fact, this method easily adapts to computer use for solving linear systems with dozens of variables. When elimination is used to solve a system of linear equations, the goal is to rewrite the system in a form to which back-substitution can be applied. To see how this works, consider the following two systems of linear equations. System of Three Linear Equations in Three Variables: (See Example 3.) x 2y 3z 9 x 3y 4 2x 5y 5z 17 Equivalent System in Row-Echelon Form: (See Example 1.) x 2y 3z 9 y 3z 5 z 2 The second system is said to be in row-echelon form, which means that it has a “stair-step” pattern with |
leading coefficients of 1. After comparing the two systems, it should be clear that it is easier to solve the system in row-echelon form, using back-substitution. Example 1 Using Back-Substitution in Row-Echelon Form Solve the system of linear equations. x 2y 3z 9 y 3z 5 z 2 Equation 1 Equation 2 Equation 3 Jeanne Drake/Tony Stone Images Solution From Equation 3, you know the value of To solve for Equation 2 to obtain y 32 5 Substitute 2 for z. z. y 1. Solve for y. y, substitute z 2 into Finally, substitute y 1 x 21 32 9 x 1. and The solution is triple 1, 1, 2. x 1, y 1, z 2 into Equation 1 to obtain Substitute 1 for y and 2 for z. Solve for x. and z 2, which can be written as the ordered Check this in the original system of equations. Now try Exercise 5. 333202_0703.qxd 12/5/05 9:42 AM Page 520 520 Chapter 7 Systems of Equations and Inequalities Historical Note One of the most influential Chinese mathematics books was the Chui-chang suan-shu or Nine Chapters on the Mathematical Art (written in approximately 250 B.C.). Chapter Eight of the Nine Chapters contained solutions of systems of linear equations using positive and negative numbers. One such system was as follows. 3x 2y z 39 2x 3y z 34 x 2y 3z 26 This system was solved using column operations on a matrix. Matrices (plural for matrix) will be discussed in the next chapter. As demonstrated in the first step in the solution of Example 2, interchanging rows is an easy way of obtaining a leading coefficient of 1. Gaussian Elimination Two systems of equations are equivalent if they have the same solution set. To solve a system that is not in row-echelon form, first convert it to an equivalent system that is in row-echelon form by using the following operations. Operations That Produce Equivalent Systems Each of the following row operations on a system of linear equations produces an equivalent system of linear equations. 1. Interchange two equations. 2. Multiply one of the equations by a nonzero constant. 3. Add a multiple of one of the equations to another equation to replace the latter equation. To see how this is done, take another look at |
the method of elimination, as applied to a system of two linear equations. Example 2 Using Gaussian Elimination to Solve a System Solve the system of linear equations. 1 0 3x 2y x y Equation 1 Equation 2 Solution There are two strategies that seem reasonable: eliminate the variable or eliminate the variable The following steps show how to use the first strategy. y. x x y 0 1 3x 2y 3x 3y 3x 2y 3x 3y 0 3x 2y Interchange the two equations in the system. Multiply the first equation by 3. Add the multiple of the first equation to the second equation to obtain a new second equation. New system in row-echelon form Now, using back-substitution, you can determine that the solution is x 1, in the original system of equations. which can be written as the ordered pair 1, 1. y 1 and Check this solution Now try Exercise 13. 333202_0703.qxd 12/5/05 9:42 AM Page 521 Arithmetic errors are often made when performing elementary row operations. You should note the operation performed in each step so that you can go back and check your work. Section 7.3 Multivariable Linear Systems 521 As shown in Example 2, rewriting a system of linear equations in row-echelon form usually involves a chain of equivalent systems, each of which is obtained by using one of the three basic row operations listed on the previous page. This process is called Gaussian elimination, after the German mathematician Carl Friedrich Gauss (1777–1855). Example 3 Using Gaussian Elimination to Solve a System Solve the system of linear equations. x 2y 3z 9 x 3y 4 2x 5y 5z 17 Equation 1 Equation 2 Equation 3 Solution Because the leading coefficient of the first equation is 1, you can begin by saving the at the upper left and eliminating the other -terms from the first column. x x x 3y x 2y 3z 9 4 y 3z 5 x 2y 3z 9 y 3z 5 2x 5y 5z 17 2x 4y 6z 18 2x 5y 5z 17 y z 1 x 2y 3z 9 y 3z 5 y z 1 Write Equation 1. Write Equation 2. Add Equation 1 to Equation 2. Adding the first equation to the second equation produces a new second equation. Multiply Equ |
ation 1 by 2. Write Equation 3. Add revised Equation 1 to Equation 3. 2 Adding times the first equation to the third equation produces a new third equation. Now that all but the first have been eliminated from the first column, go to work on the second column. (You need to eliminate from the third equation.) y x x 2y 3z 9 y 3z 5 2z 4 Adding the second equation to the third equation produces a new third equation. Finally, you need a coefficient of 1 for z in the third equation. x 2y 3z 9 y 3z 5 z 2 Multiplying the third equation 1 by produces a new third 2 equation. This is the same system that was solved in Example 1, and, as in that example, you can conclude that the solution is x 1, y 1, and z 2. Now try Exercise 15. 333202_0703.qxd 12/5/05 9:42 AM Page 522 522 Chapter 7 Systems of Equations and Inequalities The next example involves an inconsistent system—one that has no solution. The key to recognizing an inconsistent system is that at some stage in the elimination process you obtain a false statement such as 0 2. Example 4 An Inconsistent System Solve the system of linear equations. x 3y z 1 2x y 2z 2 x 2y 3z 1 Solution Equation 1 Equation 2 Equation 3 FIGURE 7.12 Solution: one point FIGURE 7.13 Solution: one line 5y 4z 0 x 2y 3z 1 x 3y z 1 x 3y z 1 x 3y z 1 5y 4z 0 5y 4z 2 5y 4z 0 0 2 2 times the first Adding equation to the second equation produces a new second equation. 1 times the first Adding equation to the third equation produces a new third equation. 1 times the second Adding equation to the third equation produces a new third equation. 0 2 is a false statement, you can conclude that this system is Because inconsistent and so has no solution. Moreover, because this system is equivalent to the original system, you can conclude that the original system also has no solution. Now try Exercise 19. FIGURE 7.14 Solution: one plane As with a system of linear equations in two variables, the solution(s) of a system of linear equations in more than two variables must fall into one of three categories. The Number of Solutions of a Linear System For a system of linear |
equations, exactly one of the following is true. FIGURE 7.15 Solution: none 1. There is exactly one solution. 2. There are infinitely many solutions. 3. There is no solution. In Section 7.2, you learned that a system of two linear equations in two variables can be represented graphically as a pair of lines that are intersecting, coincident, or parallel. A system of three linear equations in three variables has a similar graphical representation—it can be represented as three planes in space that intersect in one point (exactly one solution) [see Figure 7.12], intersect in a line or a plane (infinitely many solutions) [see Figures 7.13 and 7.14], or have no points common to all three planes (no solution) [see Figures 7.15 and 7.16]. FIGURE 7.16 Solution: none 333202_0703.qxd 12/7/05 4:22 PM Page 523 Section 7.3 Multivariable Linear Systems 523 Example 5 A System with Infinitely Many Solutions Solve the system of linear equations. x y 3z 1 y z 0 x 2y 1 Solution Equation 1 Equation 2 Equation 3 y z 0 3y 3z 0 x y 3z 1 x y 3z 1 y z 0 0 0 Adding the first equation to the third equation produces a new third equation. 3 times the second Adding equation to the third equation produces a new third equation. This result means that Equation 3 depends on Equations 1 and 2 in the sense that it gives no additional information about the variables. Because is a true statement, you can conclude that this system will have infinitely many solutions. However, it is incorrect to say simply that the solution is “infinite.” You must also specify the correct form of the solution. So, the original system is equivalent to the system 0 0 x y 3z 1. y z 0 y in the first equation produces In the last equation, solve for for a real number, the solutions to the given system are all of the form y a, So, every ordered triple of the form y z. Finally, letting x 2z 1. in terms of to obtain y z Back-substituting z a, where is x 2a 1, a and z a. 2a 1, a, a, a is a real number is a solution of the system. Now try Exercise 23. In Example 5, there are other ways to write |
the same infinite set of solutions. For instance, letting b, 1 2 b 1, 1 2 x b, b 1, the solutions could have been written as b is a real number. To convince yourself that this description produces the same set of solutions, consider the following. y and are solved x In Example 5, z. in terms of the third variable To write the correct form of the solution to the system that does not use any of the three variables a represent any of the system, let z a. real number and let Then x y. solve for and The solution can then be written in terms of a, variables of the system. which is not one of the When comparing descriptions of an infinite solution set, keep in mind that there is more than one way to describe the set. Substitution Solution 1 1 1, 0, 0 2 1 1, 1 2 20 1, 0, 0 1, 0, 0 1, 1 21 1, 1, 1 1, 1, 1 1, 1 1 1, 1 2 22 1, 2, 2 3, 2, 2 3, 1 3 1, 1 2 2 2 1 1 1, 1, 1 3 1 3, 2, 2 Same solution Same solution Same solution 333202_0703.qxd 12/5/05 9:42 AM Page 524 524 Chapter 7 Systems of Equations and Inequalities Nonsquare Systems So far, each system of linear equations you have looked at has been square, which means that the number of equations is equal to the number of variables. In a nonsquare system, the number of equations differs from the number of variables. A system of linear equations cannot have a unique solution unless there are at least as many equations as there are variables in the system. Example 6 A System with Fewer Equations than Variables Solve the system of linear equations. x 2y z 2 2x y z 1 Equation 1 Equation 2 Solution Begin by rewriting the system in row-echelon form. x 2y z 2 3y 3z 3 x 2y z 2 y z 1 2 times the first Adding equation to the second equation produces a new second equation. Multiplying the second equation 1 by produces a new second 3 equation. Solve for in terms of z, to obtain y y z 1. By back-substituting into Equation 1, you can solve for x, as follows. x 2y z 2 x 2z 1 z 2 x 2 |
z 2 z 2 x z Write Equation 1. Substitute for y in Equation 1. Distributive Property Solve for x. Finally, by letting x a, z a, y a 1, where a is a real number, you have the solution and z a. So, every ordered triple of the form a, a 1, a, a is a real number is a solution of the system. Because there were originally three variables and only two equations, the system cannot have a unique solution. Now try Exercise 27. In Example 6, try choosing some values of to obtain different solutions of the system, such as Then check each of the solutions in the original system to verify that they are solutions of the original system. 1, 0, 1, 2, 1, 2, a 3, 2, 3. and 333202_0703.qxd 12/5/05 9:42 AM Page 525 Section 7.3 Multivariable Linear Systems 525 Applications t = 1 t = 2 Example 7 Vertical Motion t The height at time of an object that is moving in a (vertical) line with constant acceleration s 1 is given by the position equation 2 at 2 v0 t s0. a t = 3 t = 0 a is measured in feet, the acceleration is the initial velocity (at t s The height is measured in seconds, squared, a, initial height. Find the values of t 3, and s 20 at v0 v0, t 1, s0 and interpret the result. (See Figure 7.17.) s 52 and if is measured in feet per second t 0), is the t 2, s 52 s0 at and at s 60 55 50 45 40 35 30 25 20 15 10 5 FIGURE 7.17 Solution s By substituting the three values of and s0 obtain three linear equations in and. 2 a1 2 v0 52 2 a22 v0 52 2 a32 v0 20 t v0, a, 1 s0 2 s0 3 s0 t 1: t 2: t 3: When When When 1 1 1 into the position equation, you can 2a 2v0 2a 2v0 9a 6v0 2s0 2s0 2s0 104 152 140 This produces the following system of linear equations. a 2a 9a 2v0 2v0 6v0 2s0 s0 2s0 104 52 40 Now solve the system using Gaussian elimination. 9a 2v0 2v |
0 6v0 2v0 2v0 12v0 a a a a 2v0 2v0 2v0 v0 2s0 3s0 2s0 2s0 3s0 16s0 2s0 3s0 2s0 2s0 3 2s0 s0 104 156 40 104 156 896 104 156 40 104 78 20 2 Adding times the first equation to the second equation produces a new second equation. 9 Adding times the first equation to the third equation produces a new third equation. 6 Adding times the second equation to the third equation produces a new third equation. 1 2 Multiplying the second equation by produces a new second equation and multiplying the third equation by produces a new third equation. 1 2 48, 20. s0 and This solution So, the solution of this system is 16t 2 48t 20 and implies that the results in a position equation of object was thrown upward at a velocity of 48 feet per second from a height of 20 feet. a 32, s v0 Now try Exercise 39. 333202_0703.qxd 12/5/05 9:42 AM Page 526 526 Chapter 7 Systems of Equations and Inequalities Example 8 Data Analysis: Curve-Fitting Find a quadratic equation y ax 2 bx c whose graph passes through the points 1, 3, 1, 1, and 2, 6. (2, 6) y = 2x2 − x y 6 5 4 3 2 (−1, 3) (1, 1) − 3 − 2 −1 1 2 3 x FIGURE 7.18 y ax 2 bx c you can write the following. passes through the points 1, 3, Solution Because the graph of 1, 1, and 2, 6, x 1, y 3: x 1, y 1: x 2, y 6: When When When a12 b1 c 3 a1 2 b1 c 1 a2 2 b2 c 6 This produces the following system of linear equations. a b c 3 a b c 1 4a 2b c 6 Equation 1 Equation 2 Equation 3 The solution of this system is parabola is y 2x 2 x, a 2, b 1, and as shown in Figure 7.18. c 0. So, the equation of the Now try Exercise 43. Example 9 Investment Analysis An inheritance of $12,000 was invested among three funds: a money-market fund that paid 5% annually, |
municipal bonds that paid 6% annually, and mutual funds that paid 12% annually. The amount invested in mutual funds was $4000 more than the amount invested in municipal bonds. The total interest earned during the first year was $1120. How much was invested in each type of fund? Solution x, y, Let represent the amounts invested in the money-market fund, municipal bonds, and mutual funds, respectively. From the given information, you can write the following equations. and z x y z 12,000 z y 4000 0.05x 0.06y 0.12z 1120 Equation 1 Equation 2 Equation 3 Rewriting this system in standard form without decimals produces the following. x 5x y y 6y z z 12z 12,000 4,000 112,000 Equation 1 Equation 2 Equation 3 Using Gaussian elimination to solve this system yields z 7000. ed in municipal bonds, and $7000 was invested in mutual funds. and So, $2000 was invested in the money-market fund, $3000 was invest- x 2000, y 3000, Now try Exercise 53. 333202_0703.qxd 12/7/05 4:23 PM Page 527 Section 7.3 Multivariable Linear Systems 527 7.3 Exercises VOCABULARY CHECK: Fill in the blanks. 1. A system of equations that is in ________ form has a “stair-step” pattern with leading coefficients of 1. 2. A solution to a system of three linear equations in three unknowns can be written as an ________ ________, which has the form x, y, z. 3. The process used to write a system of linear equations in row-echelon form is called ________ elimination. 4. Interchanging two equations of a system of linear equations is a ________ ________ that produces an equivalent system. 5. A system of equations is called ________ if the number of equations differs from the number of variables in the system. s 1 2 at2 v0t s0 6. The equation height of an object at time s t is called the ________ equation, and it models the that is moving in a vertical line with a constant acceleration a. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–4, determine whether each ordered triple |
is a solution of the system of equations. 1. 3x y z 1 2x 3z 14 5y 2z 8 (a) (c) 2, 0, 3 0, 1, 3 2. 3x 4y z 17 5x y 2z 2 2x 3y 7z 21 (b) (d) 2, 0, 8 1, 0, 4 (a) (c) 3, 1, 2 4, 1, 3 (b) (d) 1, 3, 2 1, 2, 2 7. 8. 9. 10. y z 12 z 2 3y 8z 9 z 3 2x y 3z 10 x y 2z 22 4x 2y z 8 5x y z 4 z 2 8z 5z z 22 10 4 3y 3. 4. z z 4x y 8x 6y 3x y 1 2, 3 1 2, 3 (c) (a) 4 4, 7 4, 5 4 0 7 4 9 4 (b) (d) 3 1 2, 5 2, 1 4, 5 6, 3 4 4 4x y 8z 6 y z 0 4x 7y 6 2, 2, 2 1 2, 1 8, 1 (a) (c) 2 (b) (d) 33 11 2, 10, 10 2, 4, 4 In Exercises 5–10, use back-substitution to solve the system of linear equations. 5. 2x y 5z 24 y 2z 6 z 4 6. 4x 3y 2z 21 6y 5z 8 z 2 In Exercises 11 and 12, perform the row operation and write the equivalent system. 11. Add Equation 1 to Equation 2. x x 2x 2y 3y 3z 5 5z 4 3z 0 Equation 1 Equation 2 Equation 3 What did this operation accomplish? 12. Add 2 times Equation 1 to Equation 3. x x 2x 2y 3y 3z 5z 3z 5 4 0 Equation 1 Equation 2 Equation 3 What did this operation accomplish? 333202_0703.qxd 12/5/05 9:42 AM Page 528 528 Chapter 7 Systems of Equations and Inequalities In Exercises 13–38, solve the system of linear equations and check any solution algebraically. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22 |
. 23. 24. 25. 26. 27. 1 2 1 12 9 10 7 6 0 7 9 5 x 4z 3z 2z 4z 2x y y x x 6y 3y 2x 3x z 3z z y 2y 3y 4y 2y y 5x 3y 3y x 2y 2z 3x y z 2x 4y 2z x 4y 4z 5 4z 2 4 4 x x 2x 2x 3x 2x 4y z 2x y z 5x 3y 2z 3 3x 5y 5z 1 2x y 3z 1 x 2y 7z 2x 3x 3y 6z 6 x 2z 5 3x y z 1 6x y 5z 16 x 2 0 2x y z 3x 9y 36z y 2x 6y 8z 3 6x 8y 18z 5 x 2y z 5 5x 8y 13z 7 2x 4y z 7 x 11y 4z 3 5x 2y 3z 0 7x y 3z 0 2y 4x 2x 3z 2z 13z 5z z 3y 4x 4 13 33 4 10 8 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38 3y 2z 18 5x 13y 12z 80 2x 2 z 3y 4x 9y 7 2x 3y 3z 7 4x 18y 15z 44 z y y x 2x 2x 4z z 3w w 2w w w 2w w 1 10 5 z z 4z 10z 2z 2y 3y y x y 2x 3y 3x 4y x 2y x x 2x 2y 6z 2x 3y 4x 3y 17z 0 5x 4y 22z 0 4x 2y 19z 0 12x 5y z 0 23x 4y z 0 2x y z 0 2x 6y 4z 2 3x 2y 6z x y 5z 0 0 0 4x 3y 8x 3y z 3z 4 1 3 Vertical Motion In Exercises 39– 42, an object moving vertically is at the given heights at the specified times. Find for the object. the position equation 2 at 2 v0t s0 s 1 39. At 40. At 41. At 42. At feet feet feet feet At At At At At At At At second, seconds, seconds |
, second, seconds, seconds, second, seconds, seconds, second, seconds, seconds, s 128 s 80 s 0 s 48 s 64 s 48 s 452 s 372 s 260 s 132 s 100 s 36 feet feet feet feet feet feet feet feet 333202_0703.qxd 12/5/05 9:42 AM Page 529 In Exercises 43– 46, find the equation of the parabola y ax 2 bx c that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. 43. 45. 0, 0, 2, 2, 4, 0 2, 0, 3, 1, 4, 0 44. 46. 0, 3, 1, 4, 2, 3 1, 3, 2, 2, 3, 3 In Exercises 47–50, find the equation of the circle x 2 y 2 Dx Ey F 0 that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. 47. 48. 49. 50. 0, 0, 2, 2, 4, 0 0, 0, 0, 6, 3, 3 3, 1, 2, 4, 6, 8 0, 0, 0, 2, 3, 0 51. Sports In Super Bowl I, on January 15, 1967, the Green Bay Packers defeated the Kansas City Chiefs by a score of 35 to 10. The total points scored came from 13 different scoring plays, which were a combination of touchdowns, extra-point kicks, and field goals, worth 6, 1, and 3 points respectively. The same number of touchdowns and extra point kicks were scored. There were six times as many touchdowns as field goals. How many touchdowns, extra-point kicks, and field goals were scored during the game? (Source: SuperBowl.com) 52. Sports In the 2004 Women’s NCAA Final Four the University of Connecticut Championship game, Huskies defeated the University of Tennessee Lady Volunteers by a score of 70 to 61. The Huskies won by scoring a combination of two-point baskets, three-point baskets, and one-point free throws. The number of two-point baskets was two more than the number of free throws. The number of free throws was one more than two times the number of three-point baskets. What combination of scoring accounted for the Huskies’ 70 points? (Source: National Collegiate Athletic Association |
) 53. Finance A small corporation borrowed $775,000 to expand its clothing line. Some of the money was borrowed at 8%, some at 9%, and some at 10%. How much was borrowed at each rate if the annual interest owed was $67,500 and the amount borrowed at 8% was four times the amount borrowed at 10%? 54. Finance A small corporation borrowed $800,000 to expand its line of toys. Some of the money was borrowed at 8%, some at 9%, and some at 10%. How much was borrowed at each rate if the annual interest owed was $67,000 and the amount borrowed at 8% was five times the amount borrowed at 10%? Section 7.3 Multivariable Linear Systems 529 Investment Portfolio In Exercises 55 and 56, consider an investor with a portfolio totaling $500,000 that is invested in certificates of deposit, municipal bonds, blue-chip stocks, and growth or speculative stocks. How much is invested in each type of investment? 55. The certificates of deposit pay 10% annually, and the municipal bonds pay 8% annually. Over a five-year period, the investor expects the blue-chip stocks to return 12% annually and the growth stocks to return 13% annually. The investor wants a combined annual return of 10% and also wants to have only one-fourth of the portfolio invested in stocks. 56. The certificates of deposit pay 9% annually, and the municipal bonds pay 5% annually. Over a five-year period, the investor expects the blue-chip stocks to return 12% annually and the growth stocks to return 14% annually. The investor wants a combined annual return of 10% and also wants to have only one-fourth of the portfolio invested in stocks. 57. Agriculture A mixture of 5 pounds of fertilizer A, 13 pounds of fertilizer B, and 4 pounds of fertilizer C provides the optimal nutrients for a plant. Commercial brand X contains equal parts of fertilizer B and fertilizer C. Commercial brand Y contains one part of fertilizer A and two parts of fertilizer B. Commercial brand Z contains two parts of fertilizer A, five parts of fertilizer B, and two parts of fertilizer C. How much of each fertilizer brand is needed to obtain the desired mixture? 58. Agriculture A mixture of 12 liters of chemical A, 16 liters of chemical B, and 26 liters of chemical C is required to kill a destructive crop insect. Commercial spray X contains 1, 2, and 2 parts, respectively, of these chemicals. Commercial spray Y |
contains only chemical C. Commercial spray Z contains only chemicals A and B in equal amounts. How much of each type of commercial spray is needed to get the desired mixture? 59. Coffee Mixture A coffee manufacturer sells a 10-pound package of coffee that consists of three flavors of coffee. Vanilla-flavored coffee costs $2 per pound, hazelnutflavored coffee costs $2.50 per pound, and mocha-flavored coffee costs $3 per pound. The package contains the same amount of hazelnut coffee as mocha coffee. The cost of the 10-pound package is $26. How many pounds of each type of coffee are in the package? 60. Floral Arrangements A florist is creating 10 centerpieces for a wedding. The florist can use roses that cost $2.50 each, lilies that cost $4 each, and irises that cost $2 each to make the bouquets. The customer has a budget of $300 and wants each bouquet to contain 12 flowers, with twice as many roses used as the other two types of flowers combined. How many of each type of flower should be in each centerpiece? 333202_0703.qxd 12/5/05 9:42 AM Page 530 530 Chapter 7 Systems of Equations and Inequalities 61. Advertising A health insurance company advertises on television, radio, and in the local newspaper. The marketing department has an advertising budget of $42,000 per month. A television ad costs $1000, a radio ad costs $200, and a newspaper ad costs $500. The department wants to run 60 ads per month, and have as many television ads as radio and newspaper ads combined. How many of each type of ad can the department run each month? 62. Radio You work as a disc jockey at your college radio station. You are supposed to play 32 songs within two hours. You are to choose the songs from the latest rock, dance, and pop albums. You want to play twice as many rock songs as pop songs and four more pop songs than dance songs. How many of each type of song will you play? 63. Acid Mixture A chemist needs 10 liters of a 25% acid solution. The solution is to be mixed from three solutions whose concentrations are 10%, 20%, and 50%. How many liters of each solution will satisfy each condition? (a) Use 2 liters of the 50% solution. (b) Use |
as little as possible of the 50% solution. (c) Use as much as possible of the 50% solution. 64. Acid Mixture A chemist needs 12 gallons of a 20% acid solution. The solution is to be mixed from three solutions whose concentrations are 10%, 15%, and 25%. How many gallons of each solution will satisfy each condition? (a) Use 4 gallons of the 25% solution. (b) Use as little as possible of the 25% solution. (c) Use as much as possible of the 25% solution. 65. Electrical Network Applying Kirchhoff’s Laws to the I3 and are electrical network in the figure, the currents the solution of the system I1, I2, I1 3I1 I2 2I2 2I2 I3 4I3 0 7 8 find the currents. 3Ω I1 I3 4Ω 2Ω I2 7 volts 8 volts 66. Pulley System A system of pulleys is loaded with 128t1 in the ropes and the acceleration of the 32-pound pound and 32-pound weights (see figure). The tensions and weight are found by solving the system of equations t2 a t1 t1 2t2 2a a t2 0 128 32 t1 t2 where in feet per second squared. and are measured in pounds and a is measured t2 32 lb t1 128 lb (a) Solve this system. (b) The 32-pound weight in the pulley system is replaced by a 64-pound weight. The new pulley system will be modeled by the following system of equations. t1 t1 2t2 2a a t2 0 128 64 Solve this system and use your answer for the acceleration to describe what (if anything) is happening in the pulley system..., x1, y1 Fitting a Parabola In Exercises 67–70, find the least y ax 2 bx c for the squares regression parabola, x2, y2 xn, yn by solving the followpoints ing system of linear equations for Then use a, c. the regression feature of a graphing utility to confirm the result. (If you are unfamiliar with summation notation, look at the discussion in Section 9.1 or in Appendix B at the website for this text at college.hmco.com.) and b,., nc n i1 xib n i1 i a n |
x2 i1 yi n i1 xic n i1 i b n x2 i1 n i1 i c n x2 i1 i b n x3 i1 i a n x 3 i1 i a n x 4 i1 xi yi x2 i yi 333202_0703.qxd 12/5/05 9:42 AM Page 531 67. (−2, 6) (−4, 5) y 8 6 4 2 (2, 6) (4, 2) −4 −2 2 4 69. y 12 10 8 6 (0, 0) (4, 12) (3, 6) (2, 2) −8 −6 −4 −2 8642 x x 68. y 4 2 (−1, 0) (2, 5) (1, 2) (−2, 0) −4 −2 (0, 1) 2 70. y 12 10 4 2 (0, 10) (1, 9) (2, 6) (3, 0) −8 −6 −4 8642 x x Model It 71. Data Analysis: Wildlife A wildlife management team studied the reproduction rates of deer in three tracts of a wildlife preserve. Each tract contained 5 acres. In each tract, the number of females and the that had offspring the following percent of females year, were recorded. The results are shown in the table. x, y Number, x Percent, y 100 120 140 75 68 55 (a) Use the technique demonstrated in Exercises 67–70 to set up a system of equations for the data and to find a least squares regression parabola that models the data. (b) Use a graphing utility to graph the parabola and the data in the same viewing window. (c) Use the model to create a table of estimated values y. of Compare the estimated values with the actual data. (d) Use the model to estimate the percent of females that had offspring when there were 170 females. (e) Use the model to estimate the number of females when 40% of the females had offspring. Section 7.3 Multivariable Linear Systems 531 72. Data Analysis: Stopping Distance automobile braking system, the speed and the stopping distance table. In testing a new (in miles per hour) (in feet) were recorded in the y x Speed, x Stopping distance, y 30 40 50 55 105 188 (a) Use the technique demonstrated in Exercises 67–70 |
to set up a system of equations for the data and to find a least squares regression parabola that models the data. (b) Graph the parabola and the data on the same set of axes. (c) Use the model to estimate the stopping distance when the speed is 70 miles per hour. 73. Sports In Super Bowl XXXVIII, on February 1, 2004, the New England Patriots beat the Carolina Panthers by a score of 32 to 29. The total points scored came from 16 different scoring plays, which were a combination of touchdowns, extra-point kicks, two-point conversions, and field goals, worth 6, 1, 2, and 3 points, respectively. There were four times as many touchdowns as field goals and two times as many field goals as two-point conversions. How many touchdowns, extra-point kicks, two-point conversions, and field goals were scored during the game? (Source: SuperBowl.com) 74. Sports In the 2005 Orange Bowl, the University of Southern California won the National Championship by defeating the University of Oklahoma by a score of 55 to 19. The total points scored came from 22 different scoring plays, which were a combination of touchdowns, extrapoint kicks, field goals and safeties, worth 6, 1, 3, and 2 points respectively. The same number of touchdowns and extra-point kicks were scored, and there were three times as many field goals as safeties. How many touchdowns, extra-point kicks, field goals, and safeties were scored? (Source: ESPN.com) 333202_0703.qxd 12/5/05 9:42 AM Page 532 532 Chapter 7 Systems of Equations and Inequalities and In Exercises 75–78, find values of Advanced Applications y, certain optimization problems in calculus, and a Lagrange multiplier. x, that satisfy the system. These systems arise in is called 75. 76. 77. 78. y 0 x 0 x y 10 0 2x 0 2y 0 x y 4 0 2x 2x 0 2 2y 2 0 2y 0 y x2 0 2x 1 0 2x y 100 0 Skills Review In Exercises 87–90, solve the percent problem. 87. What is 71 2% 88. 225 is what percent of 150? of 85? 89. 0.5% of what number is 400? 90. 48% of what number is 132? 91. 92. In Exercises 91–96, |
perform the operation and write the result in standard form. 7 i 4 2i 6 3i 1 6i 4 i5 2i 1 2i3 4i 6 94. 93. 95. i 1 i i 4 i 1 i 2i 8 3i Synthesis 96. True or False? the statement is true or false. Justify your answer. In Exercises 79 and 80, determine whether 79. The system x 3y 6z 2y z z 16 1 3 is in row-echelon form. 80. If a system of three linear equations is inconsistent, then its graph has no points common to all three equations. 81. Think About It Are the following two systems of equations equivalent? Give reasons for your answer. x 3y z 6 2x y 2z 1 3x 2y z 2 x 3y z 7y 4z 7y 4z 6 1 16 82. Writing When using Gaussian elimination to solve a system of linear equations, explain how you can recognize that the system has no solution. Give an example that illustrates your answer. In Exercises 83–86, find two systems of linear equations that have the ordered triple as a solution. (There are many correct answers.) 4, 1, 2 3, 1 2, 7 5, 2, 1 3 2, 4, 7 83. 85. 86. 84. 4 In Exercises 97–100, (a) determine the real zeros of and (b) sketch the graph of f f. f x x 3 x2 12x f x 8x 4 32x2 f x 2x 3 5x2 21x 36 f x 6x 3 29x2 6x 5 97. 98. 99. 100. In Exercises 101–104, use a graphing utility to construct a table of values for the equation. Then sketch the graph of the equation by hand. 101. 102. 103. 104. y 4x4 5 y 5 x1 4 y 1.90.8x 3 y 3.5x2 6 2 In Exercises 105 and 106, solve the system by elimination. 105. 106. 2x y 120 x 2y 120 6x 5y 3 10x 12y 5 107. Make a Decision To work an extended application analyzing the earnings per share for Wal-Mart Stores, Inc. from 1988 text’s website at (Data Source: Wal-Mart Stores, Inc.) college.hmco.com. to 2003, visit |
this 333202_0704.qxd 12/5/05 9:43 AM Page 533 7.4 Partial Fractions Section 7.4 Partial Fractions 533 What you should learn • Recognize partial fraction decompositions of rational expressions. • Find partial fraction decompositions of rational expressions. Why you should learn it Partial fractions can help you analyze the behavior of a rational function. For instance, in Exercise 57 on page 540, you can analyze the exhaust temperatures of a diesel engine using partial fractions. © Michael Rosenfeld/Getty Images Section A.4, shows you how to combine expressions such as 1 x 2 1 x 3 5 x 2x 3. The method of partial fractions shows you how to reverse this process. 5 x 2x 3? x 2? x 3 Introduction In this section, you will learn to write a rational expression as the sum of two or more simpler rational expressions. For example, the rational expression x 7 x2 x 6 can be written as the sum of two fractions with first-degree denominators. That is, Partial fraction decomposition x 7 x2 x 6 of. Partial fraction Partial fraction Each fraction on the right side of the equation is a partial fraction, and together they make up the partial fraction decomposition of the left side. Decomposition of 1. Divide if improper: If Nx ≥ degree of Dx, obtain Nx/Dx NxDx into Partial Fractions is an improper fraction degree of divide the denominator into the numerator to Nx Dx polynomial N1 x Dx and apply Steps 2, 3, and 4 below to the proper rational expression N1 is the remainder from the division of by xDx. Dx. Note that x N1 Nx 2. Factor the denominator: Completely factor the denominator into factors of the form px qm and ax 2 bx c where ax 2 bx cn is irreducible. 3. Linear factors: For each factor of the form px qm, the partial fracm tion decomposition must include the following sum of fractions. A1 px q A2 px q2... Am px qm 4. Quadratic factors: For each factor of the form ax 2 bx cn, n tial fraction decomposition must include the following sum of Bnx Cn B2x C2 ax 2 bx cn ax 2 bx c2 B1x C1 |
ax 2 bx c... the parfractions. 333202_0704.qxd 12/5/05 9:43 AM Page 534 534 Chapter 7 Systems of Equations and Inequalities Partial Fraction Decomposition Algebraic techniques for determining the constants in the numerators of partial fractions are demonstrated in the examples that follow. Note that the techniques vary slightly, depending on the type of factors of the denominator: linear or quadratic, distinct or repeated. Example 1 Distinct Linear Factors Write the partial fraction decomposition of x 7 x 2 x 6. Solution The expression is proper, so be sure to factor the denominator. Because x 2 x 6 x 3x 2, you should include one partial fraction with a constant numerator for each linear factor of the denominator. Write the form of the decomposition as follows Write form of decomposition. Multiplying each side of this equation by the least common denominator, x 3x 2, leads to the basic equation x 7 Ax 2 Bx 3. Basic equation Because this equation is true for all you can substitute any convenient values of x A B. and Values of that are especially that will help determine the constants x 2 convenient are ones that make the factors and equal to zero. For instance, let x x 3 x 2. x, Then 2 7 A2 2 B2 3 Te c h n o l o g y You can use a graphing utility to check graphically the decomposition found in Example 1. To do this, graph x 7 x2 x 6 y1 and y2 2 x 3 1 x 2 in the same viewing window. The graphs should be identical, as shown below. 5 A0 B5 5 5B 1 B. To solve for x 3 A, let and obtain 3 7 A3 2 B3 3 10 A5 B0 10 5A 2 A. Substitute 2 for x. Substitute 3 for x. −9 6 −6 So, the partial fraction decomposition is Check this result by combining the two partial fractions on the right side of the equation, or by using your graphing utility. Now try Exercise 15. 333202_0704.qxd 12/5/05 9:43 AM Page 535 Section 7.4 Partial Fractions 535 The next example shows how to find the partial fraction decomposition of a rational expression whose denominator has a repeated linear factor. Example 2 Repeated Linear Factors Write the partial fraction decomposition of |
x 4 2x3 6x2 20x 6 x3 2x2 x. Solution This rational expression is improper, so you should begin by dividing the numerator by the denominator to obtain x 5x2 20x 6 x3 2x2 x. Because the denominator of the remainder factors as x 3 2x 2 x xx 2 2x 1 xx 12 you should include one partial fraction with a constant numerator for each power x of and and write the form of the decomposition as follows. x 1 5x 2 20x 6 xx 12 Multiplying by the LCD, A x B x 1 xx 12, C x 12 Write form of decomposition. leads to the basic equation 5x 2 20x 6 Ax 12 Bxx 1 Cx. Basic equation Letting x 1 eliminates the B 512 201 6 A1 12 B11 1 C1 - and -terms and yields A 5 20 6 0 0 C C 9. Letting x 0 eliminates the - and B C -terms and yields 502 200 6 A0 12 B00 1 C0 6 A1 0 0 6 A. use any other value for along with the known values of x and A 6, C 9, B, x 1, At this point, you have exhausted the most convenient choices for the value of C. So, using 512 201 6 61 12 B11 1 91 31 64 2B 9 2 2B 1 B. x, so to find A and So, the partial fraction decomposition is x 6 x x 4 2x3 6x2 20x 6 x3 2x2 x 1 x 1 9 x 12. Now try Exercise 27. 333202_0704.qxd 12/5/05 9:43 AM Page 536 536 Chapter 7 Systems of Equations and Inequalities Historical Note John Bernoulli (1667–1748), a Swiss mathematician, introduced the method of partial fractions and was instrumental in the early development of calculus. Bernoulli was a professor at the University of Basel and taught many outstanding students, the most famous of whom was Leonhard Euler. The procedure used to solve for the constants in Examples 1 and 2 works well when the factors of the denominator are linear. However, when the denominator contains irreducible quadratic factors, you should use a different procedure, which involves writing the right side of the basic equation in polynomial form and equating the coefficients of like terms. Then you can use a system of equations |
to solve for the coefficients. Example 3 Distinct Linear and Quadratic Factors Write the partial fraction decomposition of 3x 2 4x 4 x 3 4x. Solution This expression is proper, so factor the denominator. Because the denominator factors as x 3 4x xx 2 4 you should include one partial fraction with a constant numerator and one partial fraction with a linear numerator and write the form of the decomposition as follows. 3x 2 4x 4 x 3 4x A x Multiplying by the LCD, Bx C x 2 4 xx 2 4, Write form of decomposition. yields the basic equation 3x 2 4x 4 Ax 2 4 Bx Cx. Basic equation Expanding this basic equation and collecting like terms produces 3x 2 4x 4 Ax 2 4A Bx 2 Cx A Bx 2 Cx 4A. Polynomial form Finally, because two polynomials are equal if and only if the coefficients of like terms are equal, you can equate the coefficients of like terms on opposite sides of the equation. 3x 2 4x 4 A Bx 2 Cx 4A Equate coefficients of like terms. You can now write the following system of linear equations. B A 4A C 3 4 4 From this system you can see that A 1 into Equation 1 yields 1 B 3 ⇒ B 2. Equation 1 Equation 2 Equation 3 A 1 and C 4. Moreover, substituting So, the partial fraction decomposition is 3x 2 4x 4 x 3 4x 1 x 2x 4 x 2 4. Now try Exercise 29. 333202_0704.qxd 12/5/05 9:43 AM Page 537 Section 7.4 Partial Fractions 537 The next example shows how to find the partial fraction decomposition of a rational expression whose denominator has a repeated quadratic factor. Example 4 Repeated Quadratic Factors Write the partial fraction decomposition of 8x 3 13x x 2 22. Solution You need to include one partial fraction with a linear numerator for each power of x 2 2. 8x 3 13x x 2 22 Ax B x 2 2 Cx D x 2 22 Write form of decomposition. Multiplying by the LCD, x 2 22, yields the basic equation 8x 3 13x Ax Bx 2 2 Cx D Basic equation Ax 3 2Ax Bx 2 2B Cx D Ax 3 Bx 2 2A Cx 2 |
B D. Polynomial form Equating coefficients of like terms on opposite sides of the equation 8x 3 0x 2 13x 0 Ax 3 Bx 2 2A Cx 2B D produces the following system of linear equations. A 2A B C 2B 8 0 13 0 A 8 D and Finally, use the values 28 C 13 C 3 20 D 0 D 0 Equation 1 Equation 2 Equation 3 Equation 4 B 0 to obtain the following. Substitute 8 for A in Equation 3. Substitute 0 for B in Equation 4. A 8, B 0, C 3, and D 0, the partial fraction decomposition So, using is 8x 3 13x x 2 22 8x x2 2 3x x 2 22. Check this result by combining the two partial fractions on the right side of the equation, or by using your graphing utility. Now try Exercise 49. 333202_0704.qxd 12/5/05 9:43 AM Page 538 538 Chapter 7 Systems of Equations and Inequalities Guidelines for Solving the Basic Equation Linear Factors 1. Substitute the zeros of the distinct linear factors into the basic equation. 2. For repeated linear factors, use the coefficients determined in Step 1 to x rewrite the basic equation. Then substitute other convenient values of and solve for the remaining coefficients. Quadratic Factors 1. Expand the basic equation. 2. Collect terms according to powers of x. 3. Equate the coefficients of like terms to obtain equations involving A, B, C, and so on. 4. Use a system of linear equations to solve for A, B, C,.... Keep in mind that for improper rational expressions such as Nx Dx 2x3 x2 7x 7 x2 x 2 you must first divide before applying partial fraction decomposition. W RITING ABOUT MATHEMATICS Error Analysis You are tutoring a student in algebra. In trying to find a partial fraction decomposition, the student writes the following. x 2 1 xx 1 x 2 1 xx 1 B A x x 1 Ax 1 xx 1 Bx xx 1 x 2 1 Ax 1 Bx Basic equation By substituting A 1 that the following. x 0 and and B 2. x 1 into the basic equation, the student concludes However, in checking this solution, the student obtains 1 x 2 x 1 1x 1 2x xx 1 x 1 xx 1 x2 1 xx 1 What |
has gone wrong? 333202_0704.qxd 12/5/05 9:43 AM Page 539 Section 7.4 Partial Fractions 539 7.4 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The process of writing a rational expression as the sum or difference of two or more simpler rational expressions is called ________ ________ ________. 2. If the degree of the numerator of a rational expression is greater than or equal to the degree of the denominator, then the fraction is called ________. 3. In order to find the partial fraction decomposition of a rational expression, the denominator must be and ________ factors of the form px qm completely factored into ________ factors of the form ax2 bx cn, which are ________ over the rationals. 4. The ________ ________ is derived after multiplying each side of the partial fraction decomposition form by the least common denominator. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, match the rational expression with the form of its decomposition. [The decompositions are labeled (a), (b), (c), and (d).] (a) (c) 1. 3 x2 x 3x 1 xx 4 3x 1 xx2 4 (b) (d) 2. 4. A x B x 4 Bx C x2 4 A x 3x 1 x2x 4 3x 1 xx2 4 In Exercises 5–14, write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. 5. 7. 9. 11. 13. 7 x 2 14x 12 x 3 10x 2 4x2 3 x 53 2x 3 x 3 10x x 1 xx2 12 6. 8. 10. 12. 14. x 2 x 2 4x 3 x2 3x 2 4x 3 11x2 6x 5 x 24 x 6 2x3 8x x 4 x23x 12 In Exercises 15–38, write the partial fraction decomposition of the rational expression. Check your result algebraically. 15. 17. 1 x 2 1 1 x 2 x 16. 18. 1 4x 2 9 3 x 2 3x 19. 21. 23. 25. 27. 29. 31. 33 |
. 35. 37. 1 2x 2 x 3 x 2 x 2 x 2 12x 12 x 3 4x 4x 2 2x 1 x 2x 1 3x x 32 x 2 1 xx 2 1 x x 3 x2 2x 2 x 2 x 4 2x 2 8 x 16x 4 1 x 2 5 x 1x 2 2x 3 20. 22. 24. 26. 28. 30. 32. 34. 36. 38 4x 3 x 2 xx 4 2x 3 x 12 6x 2 1 x 2x 12 x x 1x 2 x 1 x 6 x 3 3x2 4x 12 2x 2 x 8 x 2 42 x 1 x 3 x x 2 4x 7 x 1x 2 2x 3 In Exercises 39– 44, write the partial fraction decomposition of the improper rational expression. 39. 41. 43. x2 x x2 x 1 2x 3 x2 x 5 x2 3x 2 x 4 x 13 40. 42. 44. x2 4x x2 x 6 x 3 2x2 x 1 x2 3x 4 16x 4 2x 13 333202_0704.qxd 12/5/05 9:43 AM Page 540 540 Chapter 7 Systems of Equations and Inequalities In Exercises 45–52, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result graphically. 45. 47. 49. 51. 5 x 2x 22 2x 3 4x 2 15x 5 x 2 2x 8 46. 48. 50. 52. 3x 2 7x 2 x 3 x 4x 2 1 2xx 12 x 3 x 22x 22 x 3 x 3 x 2 x 2 Graphical Analysis In Exercises 53–56, (a) write the partial fraction decomposition of the rational function, (b) identify the graph of the rational function and the graph of each term of its decomposition, and (c) state any relationship between the vertical asymptotes of the graph of the rational function and the vertical asymptotes of the graphs of the terms of the decomposition. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 53. y x 12 xx 4 y 54. y 2(x 1)2 xx2 55. y 24x 3 x2 9 y x 4 8 −4 −8 4 −4 x 2 4 |
56. y 24x2 15x 39 x2x2 10x 26 y 12 8 4 x 4 8 –4 –4 Model It 57. Thermodynamics The magnitude of the range of exhaust temperatures (in degrees Fahrenheit) in an experimental diesel engine is approximated by the model R 20004 3x 0 < x ≤ 1 R 11 7x7 4x, Model It (co n t i n u e d ) where x is the relative load (in foot-pounds). (a) Write the partial fraction decomposition of the equation. (b) The decomposition in part (a) is the difference of two fractions. The absolute values of the terms give the expected maximum and minimum temperatures of the exhaust gases for different loads. Ymax 1st term Ymin 2nd term Write the equations for Ymax and Ymin. (c) Use a graphing utility to graph each equation from part (b) in the same viewing window. (d) Determine the expected maximum and minimum temperatures for a relative load of 0.5. Synthesis 58. Writing Describe two ways of solving for the constants in a partial fraction decomposition. True or False? the statement is true or false. Justify your answer. In Exercises 59 and 60, determine whether 59. For the rational expression x x 10x 102 the partial fraction decomposition is of the form B x 102. 60. When writing the partial fraction decomposition of the A x 10 expression x 3 x 2 x2 5x 14 denominator. the first step is to factor the In Exercises 61– 64, write the partial fraction decomposition of the rational expression. Check your result algebraically. Then assign a value to the constant a to check the result graphically. 61. 63. 1 a 2 x 2 1 ya y Skills Review 62. 64. 1 xx a 1 x 1a x In Exercises 65–70, sketch the graph of the function. 65. 67. 69. f x x2 9x 18 f x x2x 3 f x x2 x 6 x 5 66. 68. f x 2x2 9x 5 f x 1 2x 3 1 70. f x 3x 1 x2 4x 12 333202_0705.qxd 12/5/05 9:45 AM Page 541 7.5 Systems of Inequalities Section 7.5 Systems of Inequalities 541 What you should learn • Sketch the graphs of inequali- |
ties in two variables. • Solve systems of inequalities. • Use systems of inequalities in two variables to model and solve real-life problems. Why you should learn it You can use systems of inequalities in two variables to model and solve real-life problems. For instance, in Exercise 77 on page 550, you will use a system of inequalities to analyze the retail sales of prescription drugs. The Graph of an Inequality a 2x 2 3y 2 ≥ 6 3x 2y < 6 a, b b and is a solution of an inequality in and are inequalities in two The statements x variables. An ordered pair if the respectively. The graph inequality is true when and are substituted for and of an inequality is the collection of all solutions of the inequality. To sketch the graph of an inequality, begin by sketching the graph of the corresponding equation. The graph of the equation will normally separate the plane into two or more regions. In each such region, one of the following must be true. 1. All points in the region are solutions of the inequality. 2. No point in the region is a solution of the inequality. y, y x So, you can determine whether the points in an entire region satisfy the inequality by simply testing one point in the region. Sketching the Graph of an Inequality in Two Variables 1. Replace the inequality sign by an equal sign, and sketch the graph of the resulting equation. (Use a dashed line for < or > and a solid line for ≤ ≥ or.) 2. Test one point in each of the regions formed by the graph in Step 1. If the point satisfies the inequality, shade the entire region to denote that every point in the region satisfies the inequality. Example 1 Sketching the Graph of an Inequality y ≥ x2 1, To sketch the graph of y x 2 1, above the parabola the points that satisfy the inequality are those lying above (or on) the parabola. begin by graphing the corresponding equation which is a parabola, as shown in Figure 7.19. By testing a point you can see that and a point below the parabola 0, 2, 0, 0 Jon Feingersh/Masterfile y ≥ x2 − 1 y y = x2 − 1 Note that when sketching the graph of an inequality in two variables, a dashed line means all points on the line or curve are not solutions of the inequality. A solid line means all points on the line or curve are solutions of the inequality. 2 |
1 (0, 0) −2 x 2 Test point above parabola −2 Test point below parabola (0, −2) FIGURE 7.19 Now try Exercise 1. 333202_0705.qxd 12/5/05 9:45 AM Page 542 542 Chapter 7 Systems of Equations and Inequalities The inequality in Example 1 is a nonlinear inequality in two variables. Most of the following examples involve linear inequalities such as a ( and are not both zero). The graph of a linear inequality is a half-plane lying on one side of the line ax by c. ax by < c b Example 2 Sketching the Graph of a Linear Inequality Sketch the graph of each linear inequality. a. x > 2 b. y ≤ 3 Te c h n o l o g y and A graphing utility can be used to graph an inequality or a system of inequalities. For instance, to graph y ≥ x 2, y x 2 enter use the shade feature of the graphing utility to shade the correct part of the graph. You should obtain the graph below. Consult the user’s guide for your graphing utility for specific keystrokes. −10 10 −10 10 y x − y < 2 1 2 x Solution a. The graph of the corresponding equation that satisfy the inequality shown in Figure 7.20. x 2 is a vertical line. The points are those lying to the right of this line, as x > 2 b. The graph of the corresponding equation y 3 is a horizontal line. The points are those lying below (or on) this line, as that satisfy the inequality shown in Figure 7.21. y ≤ 3 x > −2 y x = −2 −4 −3 −1 2 1 −1 −2 x FIGURE 7.20 Now try Exercise 32 −1 FIGURE 7.21 x 1 2 Example 3 Sketching the Graph of a Linear Inequality Sketch the graph of x y < 2. Solution is a line, as shown in Figure The graph of the corresponding equation 7.22. Because the origin satisfies the inequality, the graph consists of the half-plane lying above the line. (Try checking a point below the line. Regardless of which point you choose, you will see that it does not satisfy the inequality.) x y 2 0, 0 x − y = 2 To graph a linear inequality, Now try Exercise 9. slope-intercept form. For instance, by writing y > x 2 it can |
help to write the inequality in x y < 2 in the form you can see that the solution points lie above the line as shown in Figure 7.22. x y 2 or y x 2, (0, 0) −1 −2 FIGURE 7.22 333202_0705.qxd 12/5/05 9:45 AM Page 543 Section 7.5 Systems of Inequalities 543 Systems of Inequalities Many practical problems in business, science, and engineering involve systems of linear inequalities. A solution of a system of inequalities in and is a point x, y that satisfies each inequality in the system. y x To sketch the graph of a system of inequalities in two variables, first sketch the graph of each individual inequality (on the same coordinate system) and then find the region that is common to every graph in the system. This region represents the solution set of the system. For systems of linear inequalities, it is helpful to find the vertices of the solution region. Example 4 Solving a System of Inequalities Sketch the graph (and label the vertices) of the solution set of the system Inequality 1 Inequality 2 Inequality 3 Solution The graphs of these inequalities are shown in Figures 7.22, 7.20, and 7.21, respectively, on page 542. The triangular region common to all three graphs can be found by superimposing the graphs on the same coordinate system, as shown in Figure 7.23. To find the vertices of the region, solve the three systems of corresponding equations obtained by taking pairs of equations representing the boundaries of the individual regions. Using different colored pencils to shade the solution of each inequality in a system will make identifying the solution of the system of inequalities easier. Vertex A: 2, 4 2 2 x y x Vertex B: 5, 3 2 x y y 3 Vertex C: x y 2, 3 2 3 y y = 3 C = (− 2, 3) y B = (5, 3 Solution set A = (−2, −4) −2 −3 −4 x = −2 −1 1 −2 −3 −4 FIGURE 7.23 Note in Figure 7.23 that the vertices of the region are represented by open dots. This means that the vertices are not solutions of the system of inequalities. Now try Exercise 35. 333202_0705.qxd 12/5/05 9:45 AM Page 544 544 Chapter 7 Systems of Equations |
and Inequalities For the triangular region shown in Figure 7.23, each point of intersection of a pair of boundary lines corresponds to a vertex. With more complicated regions, two border lines can sometimes intersect at a point that is not a vertex of the region, as shown in Figure 7.24. To keep track of which points of intersection are actually vertices of the region, you should sketch the region and refer to your sketch as you find each point of intersection. y Not a vertex x FIGURE 7.24 Example 5 Solving a System of Inequalities Sketch the region containing all points that satisfy the system of inequalities. x2 y ≤ 1 x y ≤ 1 Inequality 1 Inequality 2 Solution As shown in Figure 7.25, the points that satisfy the inequality x 2 y ≤ 1 Inequality 1 are the points lying above (or on) the parabola given by y x 2 1. Parabola The points satisfying the inequality x y ≤ 1 Inequality 2 y = x2 + 2 (−1, 0) FIGURE 7.25 (2, 3) are the points lying below (or on) the line given by y x 1. Line To find the points of intersection of the parabola and the line, solve the system of corresponding equations. x 2 x2 y 1 x y 1 and So, the region containing all points that satisfy the system is indicated by Using the method of substitution, you can find the solutions to be 2, 3. the shaded region in Figure 7.25. 1, 0 Now try Exercise 37. 333202_0705.qxd 12/5/05 9:45 AM Page 545 Section 7.5 Systems of Inequalities 545 When solving a system of inequalities, you should be aware that the system might have no solution or it might be represented by an unbounded region in the plane. These two possibilities are shown in Examples 6 and 7. Example 6 A System with No Solution Sketch the solution set of the system of inequalities. x y > x y < 3 1 Inequality 1 Inequality 2 Solution From the way the system is written, it is clear that the system has no solution, x y because the quantity and greater than 3. Graphically, the inequality is represented by the half-plane lying x y < 1 above the line is represented by the as shown in Figure 7.26. These two half-plane lying below the line half-planes have no points in common. So, the system of inequalities |
has no solution. and the inequality x y 1, cannot be both less than x y 32 −1 1 2 3 x −1 −2 x + y < −1 FIGURE 7.26 Now try Exercise 39. Example 7 An Unbounded Solution Set Sketch the solution set of the system of inequalities. x y < 3 x 2y > 3 Inequality 1 Inequality 2 Solution The graph of the inequality x y 3, x 2y 3. the half-plane that lies above the line half-planes is an infinite wedge that has a vertex at the system of inequalities is unbounded. x y < 3 is the half-plane that lies below the line is The intersection of these two 3, 0. So, the solution set of x 2y > 3 as shown in Figure 7.27. The graph of the inequality + 2y = 3 (3, 0) −1 1 2 3 x FIGURE 7.27 Now try Exercise 41. 333202_0705.qxd 12/5/05 9:45 AM Page 546 546 p Chapter 7 Systems of Equations and Inequalities Applications Consumer surplus Demand curve Equilibrium point e c i r P Supply curve Producer surplus Number of units x FIGURE 7.28 Example 9 in Section 7.2 discussed the equilibrium point for a system of demand and supply functions. The next example discusses two related concepts that economists call consumer surplus and producer surplus. As shown in Figure 7.28, the consumer surplus is defined as the area of the region that lies below the demand curve, above the horizontal line passing through the equilibrium point, and to the p -axis. Similarly, the producer surplus is defined as the area of the right of the region that lies above the supply curve, below the horizontal line passing through p the equilibrium point, and to the right of the -axis. The consumer surplus is a measure of the amount that consumers would have been willing to pay above what they actually paid, whereas the producer surplus is a measure of the amount that producers would have been willing to receive below what they actually received. Example 8 Consumer Surplus and Producer Surplus The demand and supply functions for a new type of personal digital assistant are given by p 150 0.00001x p 60 0.00002x Demand equation Supply equation 175 150 125 100 75 50 25 ) Supply vs. Demand p p = 150 − 0.00001x Consumer surplus p = 120 Producer surplus p where consumer surplus and producer surplus for these two equations. is the price |
(in dollars) and represents the number of units. Find the x Solution Begin by finding the equilibrium point (when supply and demand are equal) by solving the equation 60 0.00002x 150 0.00001x. In Example 9 in Section 7.2, you saw that the solution is p $120. which corresponds to an equilibrium price of surplus and producer surplus are the areas of the following triangular regions. units, So, the consumer x 3,000,000 p = 60 + 0.00002x 1,000,000 3,000,000 Number of units x Consumer Surplus p ≤ 150 0.00001x p ≥ 120 x ≥ 0 Producer Surplus p ≥ 60 0.00002x p ≤ 120 x ≥ 0 FIGURE 7.29 In Figure 7.29, you can see that the consumer and producer surpluses are defined as the areas of the shaded triangles. Consumer surplus Producer surplus 1 2 1 2 1 2 1 2 (base)(height) 3,000,00030 $45,000,000 (base)(height) 3,000,00060 $90,000,000 Now try Exercise 65. 333202_0705.qxd 12/5/05 9:45 AM Page 547 Section 7.5 Systems of Inequalities 547 Example 9 Nutrition The liquid portion of a diet is to provide at least 300 calories, 36 units of vitamin A, and 90 units of vitamin C. A cup of dietary drink X provides 60 calories, 12 units of vitamin A, and 10 units of vitamin C. A cup of dietary drink Y provides 60 calories, 6 units of vitamin A, and 30 units of vitamin C. Set up a system of linear inequalities that describes how many cups of each drink should be consumed each day to meet or exceed the minimum daily requirements for calories and vitamins. Solution Begin by letting and x y represent the following. x y number of cups of dietary drink X number of cups of dietary drink Y To meet or exceed the minimum daily requirements, the following inequalities must be satisfied. 60x 60y ≥ 12x 6y ≥ 10x 30y ≥ x ≥ y ≥ 300 36 90 0 0 Calories Vitamin A Vitamin C The last two inequalities are included because and cannot be negative. The graph of this system of inequalities is shown in Figure 7.30. (More is said about this application in Example 6 in Section 7.6.) y x y 8 6 4 2 (0, 6) (1, 4) ( |
3, 2) (9, 0) x 2 4 6 8 10 FIGURE 7.30 Now try Exercise 69. W RITING ABOUT MATHEMATICS Creating a System of Inequalities Plot the points coordinate plane. Draw the quadrilateral that has these four points as its vertices. Write a system of linear inequalities that has the quadrilateral as its solution. Explain how you found the system of inequalities. and in a 4, 0, 3, 2, 0, 0, 0, 2 333202_0705.qxd 12/5/05 9:45 AM Page 548 548 Chapter 7 Systems of Equations and Inequalities 7.5 Exercises VOCABULARY CHECK: Fill in the blanks. a, b 1. An ordered pair a when and are substituted for and y, b x respectively. is a ________ of an inequality in and x y if the inequality is true 2. The ________ of an inequality is the collection of all solutions of the inequality. 3. The graph of a ________ inequality is a half-plane lying on one side of the line ax by c. 4. A ________ of a system of inequalities in and x y is a point x, y that satisfies each inequality in the system. 5. The area of the region that lies below the demand curve, above the horizontal line passing through the equilibrium point, to the right of the -axis is called the ________ _________. p PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–14, sketch the graph of the inequality. 29. y 30. 1. 3. 5. 7. 9. 11. 12. y < 2 x2 2y x ≥ 4 x 12 y 22 < 9 x 12 y 42 > 9 2. 4. 6. 8. 10 > 2x 4 5x 3y ≥ 15 13. y ≤ 1 1 x 2 14. y > 15 x2 x 4 In Exercises 15–26, use a graphing utility to graph the inequality. Shade the region representing the solution. 15. 17. 19. 21. 23. 25. y < ln x y < 3x4 y ≥ 2 3x 1 y < 3.8x 1.1 x 2 5y 10 ≤ 0 5 2 y 3x2 6 ≥ 0 16. 18. 20. |
22. 24. 26. y ≥ 6 lnx 5 y ≤ 22x0.5 7 y ≤ 6 3 2x y ≥ 20.74 2.66x 2x 10 x2 3 4 In Exercises 27–30, write an inequality for the shaded region shown in the figure. 6 4 2 −4 −2 −2 2 −4 x y 4 2 −2 −4 x 2 4 In Exercises 31–34, determine whether each ordered pair is a solution of the system of linear inequalities. 31. 32. 33. 34. y > 3 y ≤ 8x 3 x ≥ 4 2x 5y ≥ 3 x2 y2 ≥ 36 y < 4 4x 2y < 7 3x y > y 1 2 x2 ≤ 15x 4y > 3x y ≤ 10 a) (c) 0, 0 4, 0 1, 3 (b) (d) 3, 11 (a) (c) 0, 2 8, 2 6, 4 (b) (d) 3, 2 (a) (c) 0, 10 2, 9 (a) (c) 1, 7 6, 0 0, 1 (b) (d) 1, 6 5, 1 (b) (d) 4, 8 27. y 4 28. y 4 2 −4 −2 −2 x 2 −4 In Exercises 35–48, sketch the graph and label the vertices of the solution set of the system of inequalities. x 4 35. 37. x y ≤ 1 x2 36. 2y 3x 38. 2x2 333202_0705.qxd 12/5/05 9:45 AM Page 549 39. 41. 43. 45. 47. x 2x 2y < 4y > y < 2x y > 2 6x 3y < 2 3x x > y2 x < y 2 x2 y2 ≤ 9 x2 y2 ≥ 1 3x 4 ≥ y2 x y < 0 6 2 3 40. 42. 44. 46. 48. 36 5 6 x 7y > 5x 2y > 6x 5y > x 2y < 6 5x 3y > 9 x y2 > 0 x y > 2 x2 ≤ 25 y2 ≤ 0 3y x < 2y y2 0 < x y 4x Section 7.5 Systems of Inequalities 549 59. y 60, 8 ) 1 2 3 4 |
x 61. Rectangle: vertices at 2, 1, 5, 1, 5, 7, 2, 7 62. Parallelogram: vertices at 0, 0, 4, 0, 1, 4, 5, 4 63. Triangle: vertices at 64. Triangle: vertices at 0, 0, 5, 0, 2, 3 1, 0, 1, 0, 0, 1 In Exercises 49–54, use a graphing utility to graph the inequalities. Shade the region representing the solution set of the system. Supply and Demand In Exercises 65–68, (a) graph the systems representing the consumer surplus and producer surplus for the supply and demand equations and (b) find the consumer surplus and producer surplus. 49. 51. 53. y ≤ 3x 1 y ≥ x2 1 y < x 3 2x 1 x2y ≥ 1 y > 2x 50. 52. 54. y > y < x2 2x 3 x2 4x 3 y ≥ x4 2x2 1 y ≤ 1 x2 y ≤ ex 22 0 x ≤ 2 y ≥ 2 ≤ In Exercises 55–64, derive a set of inequalities to describe the region. 55. y 56. y 4 3 2 1 572 2 4 8 6 4 2 −2 −2 58. 6 y 3 1 −3 −1 1 3 −3 x x x x Demand p 50 0.5x p 100 0.05x p 140 0.00002x p 400 0.0002x 65. 66. 67. 68. Supply p 0.125x p 25 0.1x p 80 0.00001x p 225 0.0005x 11 3 69. Production A furniture company can sell all the tables and chairs it produces. Each table requires 1 hour in the hours in the finishing center. Each assembly center and chair requires hours in the finishing center. The company’s assembly center is available 12 hours per day, and its finishing center is available 15 hours per day. Find and graph a system of inequalities describing all possible production levels. hours in the assembly center and 11 2 11 2 70. Inventory A store sells two models of computers. Because of the demand, the store stocks at least twice as many units of model A as of model B. The costs to the store for the two models are $800 and $1200, respectively. The management does not want more than $20,000 in computer inventory at any one time |
, and it wants at least four model A computers and two model B computers in inventory at all times. Find and graph a system of inequalities describing all possible inventory levels. 71. Investment Analysis A person plans to invest up to $20,000 in two different interest-bearing accounts. Each account is to contain at least $5000. Moreover, the amount in one account should be at least twice the amount in the other account. Find and graph a system of inequalities to describe the various amounts that can be deposited in each account. 333202_0705.qxd 12/5/05 9:45 AM Page 550 550 Chapter 7 Systems of Equations and Inequalities 72. Ticket Sales For a concert event, there are $30 reserved seat tickets and $20 general admission tickets. There are 2000 reserved seats available, and fire regulations limit the number of paid ticket holders to 3000. The promoter must take in at least $75,000 in ticket sales. Find and graph a system of inequalities describing the different numbers of tickets that can be sold. 73. Shipping A warehouse supervisor is told to ship at least 50 packages of gravel that weigh 55 pounds each and at least 40 bags of stone that weigh 70 pounds each. The maximum weight capacity in the truck he is loading is 7500 pounds. Find and graph a system of inequalities describing the numbers of bags of stone and gravel that he can send. 74. Truck Scheduling A small company that manufactures two models of exercise machines has an order for 15 units of the standard model and 16 units of the deluxe model. The company has trucks of two different sizes that can haul the products, as shown in the table. Truck Standard Deluxe Large Medium 6 4 3 6 Find and graph a system of inequalities describing the numbers of trucks of each size that are needed to deliver the order. 75. Nutrition A dietitian is asked to design a special dietary supplement using two different foods. Each ounce of food X contains 20 units of calcium, 15 units of iron, and 10 units of vitamin B. Each ounce of food Y contains 10 units of calcium, 10 units of iron, and 20 units of vitamin B. The minimum daily requirements of the diet are 300 units of calcium, 150 units of iron, and 200 units of vitamin B. (a) Write a system of inequalities describing the different amounts of food X and food Y that can be used. (b) Sketch a graph of the region corresponding to the system in part (a). (c) Find two solutions of the system |
and interpret their meanings in the context of the problem. x 76. Health A person’s maximum heart rate is is the person’s age in years for 220 x, 20 ≤ x ≤ 70. where it is recommended that the When a person exercises, person strive for a heart rate that is at least 50% of the maximum and at most 75% of the maximum. (Source: American Heart Association) (a) Write a system of inequalities that describes the exercise target heart rate region. (b) Sketch a graph of the region in part (a). (c) Find two solutions to the system and interpret their meanings in the context of the problem. Model It 77. Data Analysis: Prescription Drugs The table shows the retail sales (in billions of dollars) of prescription drugs in the United States from 1999 to 2003. (Source: National Association of Chain Drug Stores) y Year Retail sales, y 1999 2000 2001 2002 2003 125.8 145.6 164.1 182.7 203.1 (a) Use the regression feature of a graphing utility to find a linear model for the data. Let represent the year, with corresponding to 1999. t 9 t (b) The total retail sales of prescription drugs in the United States during this five-year period can be approximated by finding the area of the trapezoid bounded by the linear model you found in part (a) y 0, t 8.5, and the lines Use a graphing utility to graph this region. t 13.5. and (c) Use the formula for the area of a trapezoid to approximate the total retail sales of prescription drugs. 78. Physical Fitness Facility An indoor running track is to be constructed with a space for body-building equipment inside the track (see figure). The track must be at least 125 meters long, and the body-building space must have an area of at least 500 square meters. y Body-building equipment x (a) Find a system of inequalities describing the require- ments of the facility. (b) Graph the system from part (a). 333202_0705.qxd 12/5/05 9:45 AM Page 551 Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 79 and 80, determine whether 79. The area of the figure defined by the system is 99 square units. 80. The graph below shows the solution of the system y ≤ 6 4x 9y > 6 3 |
x y2 ≥ 2. y 10 8 4 −4 −6 −8 −4 81. Writing Explain the difference between the graphs of the on the real number line and on the x ≤ 4 inequality rectangular coordinate system. 82. Think About It After graphing the boundary of an inequality in and how do you decide on which side of the boundary the solution set of the inequality lies? y, x 83. Graphical Reasoning Two concentric circles have radii The area between the circles must be y, and where x at least 10 square units. y > x. (a) Find a system of inequalities describing the constraints on the circles. (b) Use a graphing utility to graph the system of inequaliin the same y x ties in part (a). Graph the line viewing window. (c) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in the context of the problem. 84. The graph of the solution of the inequality is shown in the figure. Describe how the solution set would change for each of the following. x 2y < 6 (a) x 2y ≤ 6 (b) x 2y > 6 y 6 2 −2 −4 2 4 6 x x 6 −6 −6 Section 7.5 Systems of Inequalities 551 In Exercises 85–88, match the system of inequalities with the graph of its solution. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) y 2 −6 −2 2 −6 y 2 −6 −2 2 (b) (d) y 2 −6 −2 2 −6 y 2 −6 −2 2 x x x x 85. 87. x2 y2 ≤ 16 x y ≥ 4 x2 y2 ≥ 16 x y ≥ 4 Skills Review 86. 88. x2 y2 ≤ 16 x y ≤ 4 x2 y2 ≥ 16 x y ≤ 4 In Exercises 89–94, find the equation of the line passing through the two points. 89. 91. 93. 2, 6, 4, 4 3 7 4, 2, 2, 5 3.4, 5.2, 2.6, 0.8 90. 92. 94. 8, 0, 3, 1 1 11 2, 0, 2, 12 4.1, 3.8, 2.9, 8.2 95. Data Analysis: Cell Phone Bills The |
average monthly (in dollars) in the United States from is the year, are shown as data points (Source: Cellular Telecommunications & Internet y t cell phone bills 1998 to 2003, where t, y. Association) 1998, 39.43, 2001, 47.37, 1999, 41.24, 2002, 48.40, 2000, 45.27 2003, 49.91 (a) Use the regression feature of a graphing utility to find a linear model, a quadratic model, and an exponential t 8 model for the data. Let correspond to 1998. (b) Use a graphing utility to plot the data and the models in the same viewing window. (c) Which model is the best fit for the data? (d) Use the model from part (c) to predict the average monthly cell phone bill in 2008. 333202_0706.qxd 12/5/05 9:46 AM Page 552 552 Chapter 7 Systems of Equations and Inequalities 7.6 Linear Programming What you should learn • Solve linear programming problems. • Use linear programming to model and solve real-life problems. Why you should learn it Linear programming is often useful in making real-life economic decisions. For example, Exercise 44 on page 560 shows how you can determine the optimal cost of a blend of gasoline and compare it with the national average. Linear Programming: A Graphical Approach Many applications in business and economics involve a process called optimization, in which you are asked to find the minimum or maximum of a quantity. In this section, you will study an optimization strategy called linear programming. A two-dimensional linear programming problem consists of a linear objective function and a system of linear inequalities called constraints. The objective function gives the quantity that is to be maximized (or minimized), and the constraints determine the set of feasible solutions. For example, suppose you are asked to maximize the value of z ax by Objective function subject to a set of constraints that determines the shaded region in Figure 7.31. y Feasible solutions x FIGURE 7.31 Tim Boyle/Getty Images Because every point in the shaded region satisfies each constraint, it is not clear how you should find the point that yields a maximum value of Fortunately, it can be shown that if there is an optimal solution, it must occur at one of the vertices. This means that you can find the maximum value of z by testing z at each of the vertices. z. Optimal Solution of a Linear Programming Problem If |
a linear programming problem has a solution, it must occur at a vertex of the set of feasible solutions. If there is more than one solution, at least one of them must occur at such a vertex. In either case, the value of the objective function is unique. Some guidelines for solving a linear programming problem in two variables are listed at the top of the next page. 333202_0706.qxd 12/5/05 9:46 AM Page 553 Section 7.6 Linear Programming 553 Solving a Linear Programming Problem 1. Sketch the region corresponding to the system of constraints. (The points inside or on the boundary of the region are feasible solutions.) 2. Find the vertices of the region. 3. Test the objective function at each of the vertices and select the values of the variables that optimize the objective function. For a bounded region, both a minimum and a maximum value will exist. (For an unbounded region, if an optimal solution exists, it will occur at a vertex.) Example 1 Solving a Linear Programming Problem Find the maximum value of z 3x 2y subject to the following constraints. x ≥ 0 y ≥ 0 x 2y ≤ 4 x y ≤ 1 Objective function Constraints Solution The constraints form the region shown in Figure 7.32. At the four vertices of this region, the objective function has the following values. z 30 20 0 z 31 20 3 z 32 21 8 z 30 22 4 0, 0: 1, 0: 2, 1: 0, 2: At At At At Maximum value of z So, the maximum value of z is 8, and this occurs when x 2 and y 1. Now try Exercise 5. In Example 1, try testing some of the interior points in the region. You will see that the corresponding values of are less than 8. Here are some examples. At 1, 1: z z 31 21 5 At 1 2, 3 2 : z 31 2 23 2 9 2 To see why the maximum value of the objective function in Example 1 must occur at a vertex, consider writing the objective function in slope-intercept form y 3 2 x z 2 Family of lines z2 y 3 2. is the -intercept of the objective function. This equation represents a where family of lines, each of slope Of these infinitely many lines, you want the one that has the largest -value while still intersecting the region determined by the constraints. In other words, of all the lines whose slope is you want the one |
that has the largest -intercept and intersects the given region, as shown in Figure 7.33. From the graph you can see that such a line will pass through one (or more) of the vertices of the region. 3 2, y z y 4 3 2 1 (0, 2) x = 0 x + 2y = 4 (2, 1) x − y = 1 (1, 0) (0, 0) y = 0 2 3 FIGURE 7.32 FIGURE 7.33 x x 333202_0706.qxd 12/5/05 9:46 AM Page 554 554 Chapter 7 Systems of Equations and Inequalities (1, 5) (0, 4) (0, 2) y 5 4 3 2 1 (6, 3) (3, 0) (5, 0) x 1 2 3 4 5 6 FIGURE 7.34. Historical Note George Dantzig (1914 – ) was the first to propose the simplex method, or linear programming, in 1947. This technique defined the steps needed to find the optimal solution to a complex multivariable problem. The next example shows that the same basic procedure can be used to solve a problem in which the objective function is to be minimized. Example 2 Minimizing an Objective Function Find the minimum value of z 5x 7y Objective function x ≥ 0 where 2x 3x x 2x and 3y ≥ y ≤ y ≤ 5y ≤ y ≥ 0, 6 15 4 27 subject to the following constraints. Constraints Solution The region bounded by the constraints is shown in Figure 7.34. By testing the objective function at each vertex, you obtain the following. At At At At At At 0, 2: 0, 4: 1, 5: 6, 3: 5, 0: 3, 0: Minimum value of z z 50 72 14 z 50 74 28 z 51 75 40 z 56 73 51 z 55 70 25 z 53 70 15 So, the minimum value of z is 14, and this occurs when x 0 and y 2. Now try Exercise 13. Example 3 Maximizing an Objective Function Find the maximum value of z 5x 7y Objective function where x ≥ 0 y ≥ 0, and 3y ≥ 6 y ≤ 15 y ≤ 4 5y ≤ 27 2x 3x x 2x subject to the following constraints. Constraints Solution This linear programming problem is identical to that given in Example 2 above, except that the objective |
function is maximized instead of minimized. Using the values of at the vertices shown above, you can conclude that the maximum z value of z is z 56 73 51 and occurs when x 6 and y 3. Now try Exercise 15. 333202_0706.qxd 12/5/05 9:46 AM Page 555 y (0, 4) (2, 4) 4 3 2 1 z =12 for any point along this line segment. (5, 1) (0, 0) (5, 0) x 1 2 3 4 5 FIGURE 7.35 Section 7.6 Linear Programming 555 It is possible for the maximum (or minimum) value in a linear programming problem to occur at two different vertices. For instance, at the vertices of the region shown in Figure 7.35, the objective function z 2x 2y has the following values. At At At At At 0, 0: 0, 4: 2, 4: 5, 1: 5, 0: z 20 20 10 z 20 24 18 z 22 24 12 z 25 21 12 z 25 20 10 Objective function Maximum value of z Maximum value of z and 2, 4 5, 1; In this case, you can conclude that the objective function has a maximum value not only at the vertices it also has a maximum value (of 12) at any point on the line segment connecting these two vertices. Note that the objective function in slope-intercept form has the same slope as the line through the vertices 5, 1. Some linear programming problems have no optimal solutions. This can occur if the region determined by the constraints is unbounded. Example 4 illustrates such a problem. y x 1 2 z 2, 4 and Example 4 An Unbounded Region Find the maximum value of z 4x 2y Objective function where y ≥ 0, x ≥ 0 and x 2y ≥ 4 3x y ≥ 7 x 2y ≤ 7 subject to the following constraints. Constraints y 5 4 3 2 1 (1, 4) (2, 1) (4, 0) x 1 2 3 4 5 FIGURE 7.36 Solution The region determined by the constraints is shown in Figure 7.36. For this unbounded region, there is no maximum value of To see this, note that the point Substituting this point into the objective function, you get z 4x 20 4x. lies in the region for all values of z. x ≥ 4. x, 0 x to be large |
, you can obtain values of By choosing that are as large as you want. So, there is no maximum value of However, there is a minimum value of z. z. z At At At 1, 4: 2, 1: 4, 0: z 41 24 12 z 42 21 10 z 44 20 16 Minimum value of z So, the minimum value of z is 10, and this occurs when x 2 and y 1. Now try Exercise 17. 333202_0706.qxd 12/5/05 9:46 AM Page 556 556 Chapter 7 Systems of Equations and Inequalities Applications Example 5 shows how linear programming can be used to find the maximum profit in a business application. Example 5 Optimal Profit A candy manufacturer wants to maximize the profit for two types of boxed chocolates. A box of chocolate covered creams yields a profit of $1.50 per box, and a box of chocolate covered nuts yields a profit of $2.00 per box. Market tests and available resources have indicated the following constraints. 1. The combined production level should not exceed 1200 boxes per month. 2. The demand for a box of chocolate covered nuts is no more than half the demand for a box of chocolate covered creams. 3. The production level for chocolate covered creams should be less than or equal to 600 boxes plus three times the production level for chocolate covered nuts. Solution x Let be the be the number of boxes of chocolate covered creams and let number of boxes of chocolate covered nuts. So, the objective function (for the combined profit) is given by y P 1.5x 2y. Objective function The three constraints translate into the following linear inequalities. 1. 2. 3. x y ≤ 1200 y ≤ 1 2x x ≤ 600 3y x y ≤ 1200 x 2y ≤ 0 x 3y ≤ 600 Maximum Profit y (800, 400) (1050, 150) (0, 0) (600, 0) 400 800 1200 Boxes of chocolate covered creams 400 300 200 100 FIGURE 7.37 x x ≥ 0 can be negative, you also have the two additional Because neither constraints of Figure 7.37 shows the region determined by the constraints. To find the maximum profit, test the values of at the vertices of the region. y nor y ≥ 0. and P At 0, 0: P 1.50 0 At 800, 400: P 1.5800 2400 2000 At 1050, 150: P 1.51050 21 |
50 1875 900 At 600, 0: P 1.5600 20 20 Maximum profit x So, the maximum profit is $2000, and it occurs when the monthly production consists of 800 boxes of chocolate covered creams and 400 boxes of chocolate covered nuts. Now try Exercise 39. In Example 5, if the manufacturer improved the production of chocolate covered creams so that they yielded a profit of $2.50 per unit, the maximum profit could then be found using the objective function By testing the values of at the vertices of the region, you would find that the maximum profit was $2925 and that it occurred when P 2.5x 2y. and y 150. x 1050 P 333202_0706.qxd 12/5/05 9:46 AM Page 557 Section 7.6 Linear Programming 557 Example 6 Optimal Cost The liquid portion of a diet is to provide at least 300 calories, 36 units of vitamin A, and 90 units of vitamin C. A cup of dietary drink X costs $0.12 and provides 60 calories, 12 units of vitamin A, and 10 units of vitamin C. A cup of dietary drink Y costs $0.15 and provides 60 calories, 6 units of vitamin A, and 30 units of vitamin C. How many cups of each drink should be consumed each day to obtain an optimal cost and still meet the daily requirements? Solution As in Example 9 in Section 7.5, let be the number of cups of dietary drink X and let be the number of cups of dietary drink Y. y x For calories: For vitamin A: For vitamin C: 60x 60y ≥ 300 12x 6y ≥ 36 10x 30y ≥ 90 x ≥ 0 y ≥ 0 Constraints y 8 6 4 2 (0, 6) (1, 4) (3, 2) (9, 0) x 2 4 6 8 10 FIGURE 7.38 The cost C is given by C 0.12x 0.15y. Objective function The graph of the region corresponding to the constraints is shown in Figure 7.38. Because you want to incur as little cost as possible, you want to determine the minimum cost. To determine the minimum cost, test at each vertex of the region. C At At At At 0, 6: 1, 4: 3, 2: 9, 0: C 0.120 0.156 0.90 C 0.121 0.154 0.72 C 0.123 0.152 0.66 C |
0.129 0.150 1.08 Minimum value of C So, the minimum cost is $0.66 per day, and this occurs when 3 cups of drink X and 2 cups of drink Y are consumed each day. Now try Exercise 43. W RITING ABOUT MATHEMATICS Creating a Linear Programming Problem Sketch the region determined by the following constraints. x 2y ≤ Constraints Find, if possible, an objective function of the form maximum at each indicated vertex of the region. z ax by that has a a. 0, 4 b. 2, 3 c. 5, 0 d. 0, 0 Explain how you found each objective function. 333202_0706.qxd 12/5/05 9:46 AM Page 558 558 Chapter 7 Systems of Equations and Inequalities 7.6 Exercises VOCABULARY CHECK: Fill in the blanks. 1. In the process called ________, you are asked to find the maximum or minimum value of a quantity. 2. One type of optimization strategy is called ________ ________. 3. The ________ function of a linear programming problem gives the quantity that is to be maximized or minimized. 4. The ________ of a linear programming problem determine the set of ________ ________. 5. If a linear programming problem has a solution, it must occur at a ________ of the set of feasible solutions. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–12, find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. (For each exercise, the graph of the region determined by the constraints is provided.) 1. Objective function: z 4x 3y Constraints0, 5) y 6 5 4 3 2 1 2. Objective function: z 2x 8y Constraints: x ≥ 0 y ≥ 0 2x y ≤ 4 y (0, 4) 4 3 2 (0, 0) (5, 0) 1 2 3 4 5 6 x (0, 0) −1 (2, 0) x 1 2 3 3. Objective function: z 3x 8y Constraints: (See Exercise 1.) 5. Objective function: z 3x 2y Constraints: x ≥ 0 y ≥ 0 x 3y ≤ 15 4x y ≤ 16 4. Objective function |
: z 7x 3y Constraints: (See Exercise 2.) 6. Objective function: z 4x 5y Constraints: x ≥ 0 2x 3y ≥ 6 3x y ≤ 9 x 4y ≤ 16 y 5 4 3 2 1 (0, 5) (3, 4) (0, 0) (4, 00, 4) (0, 2) (4, 3) (3, 0) x 1 2 3 4 5 FIGURE FOR 5 FIGURE FOR 6 7. Objective function: z 5x 0.5y Constraints: (See Exercise 5.) 9. Objective function: z 10x 7y Constraints: 0 ≤ x ≤ 60 0 ≤ y ≤ 45 5x 6y ≤ 420 8. Objective function: z 2x y Constraints: (See Exercise 6.) 10. Objective function: z 25x 35y Constraints: x ≥ y ≥ 0 8x 9y ≤ 7200 8x 9y ≥ 3600 0 y 60 40 20 (0, 45) (30, 45) (60, 20) (0, 0) (60, 0) 20 40 60 x y 800 400 (0, 800) (0, 400) (900, 0) x 400 (450, 0) 11. Objective function: z 25x 30y Constraints: (See Exercise 9.) 12. Objective function: z 15x 20y Constraints: (See Exercise 10.) 333202_0706.qxd 12/5/05 9:46 AM Page 559 In Exercises 13–20, sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. 13. Objective function: z 6x 10y Constraints: x x ≥ 0 y ≥ 0 x 2x 5y ≤ 10 15. Objective function: z 9x 24y Constraints: (See Exercise 13.) 17. Objective function: z 4x 5y Constraints: x ≥ 0 y ≥ 0 x 5y ≥ 8 3x 5y ≥ 30 19. Objective function: z 2x 7y Constraints: (See Exercise 17.) x 14. Objective function: z 7x 8y Constraints 16. Objective function: z 7x 2y Constraints: (See Exercise 14.) x x 1 18. Objective function: z 4x 5y Constraints: |
x ≥ 0 y ≥ 0 2x 2y ≤ 10 x 2y ≤ 6 20. Objective function: z 2x y Constraints: (See Exercise 18.) In Exercises 21–24, use a graphing utility to graph the region determined by the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the constraints. 21. Objective function: 22. Objective function: z 4x y Constraints: x ≥ 0 y ≥ 0 x 2y ≤ 40 2x 3y ≥ 72 23. Objective function: z x 4y Constraints: (See Exercise 21.) z x Constraints: x ≥ 0 y ≥ 0 2x 3y ≤ 60 2x y ≤ 28 4x y ≤ 48 24. Objective function: z y Constraints: (See Exercise 22.) Section 7.6 Linear Programming 559 In Exercises 25–28, find the maximum value of the objective function and where it occurs, subject to the constraints x ≥ 0, and 4x 3y ≤ 30. 3x y ≤ 15, y ≥ 0, 25. 26. 27. 28. z 2x y z 5x y z x y z 3x y In Exercises 29–32, find the maximum value of the objective the function and where constraints and 2x 2y ≤ 21. to x y ≤ 18, x 4y ≤ 20, it occurs, subject x ≥ 0, y ≥ 0, 29. 30. 31. 32. z x 5y z 2x 4y z 4x 5y z 4x y In Exercises 33–38, the linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the maximum value of the objective function and where it occurs. 33. Objective function: z 2.5x y Constraints: x ≥ 0 y ≥ 0 3x 5y ≤ 15 5x 2y ≤ 10 35. Objective function: z x 2y Constraints: x ≥ 0 y ≥ 0 x ≤ 10 x y ≤ 7 37. Objective function: z 3x 4y Constraints 2x y ≤ 4 34. Objective function: z x y Constraints 2y ≤ 4 36. Objective function: z x y Constraints 3x y ≥ 3 38. Objective function: z x 2y Constraints: x ≥ 0 y ≥ 0 x 2y ≤ 4 2x |
2y ≤ 4 333202_0706.qxd 12/5/05 9:46 AM Page 560 560 Chapter 7 Systems of Equations and Inequalities 39. Optimal Profit A manufacturer produces two models of bicycles. The times (in hours) required for assembling, painting, and packaging each model are shown in the table. Process Hours, model A Hours, model B Assembling Painting Packaging 2 4 1 2.5 1 0.75 The total times available for assembling, painting, and packaging are 4000 hours, 4800 hours, and 1500 hours, respectively. The profits per unit are $45 for model A and $50 for model B. What is the optimal production level for each model? What is the optimal profit? 40. Optimal Profit A manufacturer produces two models of bicycles. The times (in hours) required for assembling, painting, and packaging each model are shown in the table. Process Assembling Painting Packaging Hours, model A Hours, model B 2.5 2 0.75 3 1 1.25 The total times available for assembling, painting, and packaging are 4000 hours, 2500 hours, and 1500 hours, respectively. The profits per unit are $50 for model A and $52 for model B. What is the optimal production level for each model? What is the optimal profit? 41. Optimal Profit A merchant plans to sell two models of MP3 players at costs of $250 and $300. The $250 model yields a profit of $25 per unit and the $300 model yields a profit of $40 per unit. The merchant estimates that the total monthly demand will not exceed 250 units. The merchant does not want to invest more than $65,000 in inventory for these products. What is the optimal inventory level for each model? What is the optimal profit? 42. Optimal Profit A fruit grower has 150 acres of land available to raise two crops, A and B. It takes 1 day to trim an acre of crop A and 2 days to trim an acre of crop B, and there are 240 days per year available for trimming. It takes 0.3 day to pick an acre of crop A and 0.1 day to pick an acre of crop B, and there are 30 days available for picking. The profit is $140 per acre for crop A and $235 per acre for crop B. What is the optimal acreage for each fruit? What is the optimal profit? 43. Optimal Cost A farming cooperative mixes two brands of |
cattle feed. Brand X costs $25 per bag and contains two units of nutritional element A, two units of element B, and two units of element C. Brand Y costs $20 per bag and contains one unit of nutritional element A, nine units of element B, and three units of element C. The minimum requirements of nutrients A, B, and C are 12 units, 36 units, and 24 units, respectively. What is the optimal number of bags of each brand that should be mixed? What is the optimal cost? Model It 44. Optimal Cost According to AAA (Automobile Association of America), on January 24, 2005, the national average price per gallon for regular unleaded (87-octane) gasoline was $1.84, and the price for premium unleaded (93-octane) gasoline was $2.03. (a) Write an objective function that models the cost of the blend of mid-grade unleaded gasoline (89octane). (b) Determine the constraints for the objective function in part (a). (c) Sketch a graph of the region determined by the constraints from part (b). (d) Determine the blend of regular and premium unleaded gasoline that results in an optimal cost of mid-grade unleaded gasoline. (e) What is the optimal cost? (f) Is the cost lower than the national average of $1.96 per gallon for mid-grade unleaded gasoline? 45. Optimal Revenue An accounting firm has 900 hours of staff time and 155 hours of reviewing time available each week. The firm charges $2500 for an audit and $350 for a tax return. Each audit requires 75 hours of staff time and 10 hours of review time. Each tax return requires 12.5 hours of staff time and 2.5 hours of review time. What numbers of audits and tax returns will yield an optimal revenue? What is the optimal revenue? 46. Optimal Revenue The accounting firm in Exercise 45 lowers its charge for an audit to $2000. What numbers of audits and tax returns will yield an optimal revenue? What is the optimal revenue? 333202_0706.qxd 12/5/05 9:46 AM Page 561 47. Investment Portfolio An investor has up to $250,000 to invest in two types of investments. Type A pays 8% annually and type B pays 10% annually. To have a well-balanced portfolio, the investor imposes the following conditions. At least one-fourth of the total portfolio is to |
be allocated to type A investments and at least one-fourth of the portfolio is to be allocated to type B investments. What is the optimal amount that should be invested in each type of investment? What is the optimal return? 48. Investment Portfolio An investor has up to $450,000 to invest in two types of investments. Type A pays 6% annually and type B pays 10% annually. To have a well-balanced portfolio, the investor imposes the following conditions. At least one-half of the total portfolio is to be allocated to type A investments and at least one-fourth of the portfolio is to be allocated to type B investments. What is the optimal amount that should be invested in each type of investment? What is the optimal return? Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 49 and 50, determine whether 49. If an objective function has a maximum value at the you can conclude that it also has 8, 3, 4, 7 vertices and a maximum value at the points 4.5, 6.5 and 7.8, 3.2. 50. When solving a linear programming problem, if the objective function has a maximum value at more than one vertex, you can assume that there are an infinite number of points that will produce the maximum value. In Exercises 51 and 52, determine values of t such that the objective function has maximum values at the indicated vertices. 51. Objective function: z 3x t y 52. Objective function: z 3x t y Constraints: (b) x ≥ 0 y ≥ 0 x 3y ≤ 15 4x y ≤ 16 0, 5 (a) 3, 4 Constraints: x ≥ 0 y ≥ 0 x 2y ≤ 4 x y ≤ 1 2, 1 (a) 0, 2 (b) Section 7.6 Linear Programming 561 Think About It In Exercises 53–56, find an objective function that has a maximum or minimum value at the indicated vertex of the constraint region shown below. (There are many correct answers.) y 6 5 3 2 1 A(0, 4) B(4, 3) C(5, 0) −1 1 432 6 x 53. The maximum occurs at vertex A. 54. The maximum occurs at vertex B. 55. The maximum occurs at vertex C. 56. The minimum occurs at vertex C. Skills Review In Exercises 57–60, simplify the complex fraction. |
57. 59 58. 60 2x 2 4x 2 1 2 In Exercises 61–66, solve the equation algebraically. Round the result to three decimal places. 61. 62. 63. 64. 65. 66. e 2x 2e x 15 0 e 2x 10e x 24 0 862 e x4 192 75 150 e x 4 7 ln 3x 12 lnx 92 2 In Exercises 67 and 68, solve the system of linear equations and check any solution algebraically. 67. 2x 6y z 17 5y z 8 x 2y 3z 23 68. 7x 3y 5z 28 4x 4z 16 7x 2y z 0 333202_070R.qxd 12/5/05 9:48 AM Page 562 562 Chapter 7 Systems of Equations and Inequalities 7 Chapter Summary What did you learn? Section 7.1 Use the method of substitution to solve systems of linear equations Review Exercises 1– 4 in two variables (p. 496). Use the method of substitution to solve systems of nonlinear equations in two variables (p. 499). Use a graphical approach to solve systems of equations in two variables (p. 500). Use systems of equations to model and solve real-life problems (p. 501). Section 7.2 Use the method of elimination to solve systems of linear equations in two variables (p.507). Interpret graphically the numbers of solutions of systems of linear equations in two variables (p. 510). Use systems of linear equations in two variables to model and solve real-life problems (p. 513). Section 7.3 Use back-substitution to solve linear systems in row-echelon form (p. 519). Use Gaussian elimination to solve systems of linear equations (p. 520). Solve nonsquare systems of linear equations (p. 524). Use systems of linear equations in three or more variables to model and solve real-life problems (p. 525). Section 7.4 Recognize partial fraction decompositions of rational expressions (p. 533). Find partial fraction decompositions of rational expressions (p. 534). Section 7.5 Sketch the graphs of inequalities in two variables (p. 541). Solve systems of inequalities (p. 543). Use systems of inequalities in two variables to model and solve real-life problems (p. 546). Section 7.6 Solve linear programming problems (p |
. 552). Use linear programming to model and solve real-life problems (p. 556). 5–8 9–14 15–18 19–26 27–30 31, 32 33, 34 35–38 39, 40 41– 48 49–52 53–60 61–64 65–72 73–76 77–82 83–86 333202_070R.qxd 12/5/05 9:48 AM Page 563 7 Review Exercises In Exercises 1–8, solve the system by the method of 7.1 substitution. In Exercises 19–26, solve the system by the method 7.2 of elimination. Review Exercises 563 1. 3. 5. 7. 1.25x y 4.5y x y 2 x y 0 0.5x x2 y2 9 x y 1 y 2x2 y x 4 2x2 0.75 2.5 2. 4. 6. 8. x 3 0 3 5 4 5 2x 3y y x 2 5y 1 x 5y x2 y2 169 3x 2y 39 x y 3 x y2 1 In Exercises 9–12, solve the system graphically. 9. 11. 10 6 2x y x 5y y y 2x2 4x 1 x2 4x 3 10. 12. 2x 3y 5y 3 28 8x y2 2y x 0 x y 0 In Exercises 13 and 14, use a graphing utility to solve the system of equations. Find the solution accurate to two decimal places. 13. 14. 2e x y y y 2ex 0 lnx 1 3 4 1 2 x y 15. Break-Even Analysis You set up a scrapbook business and make an initial investment of $50,000. The unit cost of a scrapbook kit is $12 and the selling price is $25. How many kits must you sell to break even? 16. Choice of Two Jobs You are offered two sales jobs at a pharmaceutical company. One company offers an annual salary of $35,000 plus a year-end bonus of 1.5% of your total sales. The other company offers an annual salary of $32,000 plus a year-end bonus of 2% of your total sales. What amount of sales will make the second offer better? Explain. 17. Geometry The perimeter of a rectangle is 480 meters and its length is 150% of its width. |
Find the dimensions of the rectangle. 18. Geometry The perimeter of a rectangle is 68 feet and its times its length. Find the dimensions of the 8 width is 9 rectangle. 19. 21. 23. 25. 2x y 2 6x 8y 39 0.2x 0.3y 0.14 0.4x 0.5y 0.20 3x 2y 0 3x 2 y 5 10 1.25x 2y 8y 3.5 14 5x 20. 22. 24. 26. 40x 30y 24 14 20x 50y 12x 42y 17 30x 18y 19 7x 12y 63 2x 3y 2 21 1.5x 2.5y 8.5 6x 10y 24 In Exercises 27–30, match the system of linear equations with its graph. Describe the number of solutions and state whether the system is consistent or inconsistent. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) −4 y 4 −2 −4 y 2 −2 −2 −4 −6 x 2 4 x 4 6 (b) y 4 −4 −4 (d) y 4 2 −4 x 4 x 6 27. 29. x 5y 4 x 3y 6 3x y 7 6x 2y 8 28. 30. 3x y 7 9x 3y 21 2x y 3 x 5y 4 Supply and Demand equilibrium point of demand and supply equations. In Exercises 31 and 32, find the Demand p 37 0.0002x p 120 0.0001x 31. 32. Supply p 22 0.00001x p 45 0.0002x 333202_070R.qxd 12/5/05 9:48 AM Page 564 564 Chapter 7 Systems of Equations and Inequalities In Exercises 33 and 34, use back-substitution to solve 7.3 the system of linear equations. In Exercises 43 and 44, find the equation of the circle x 2 y 2 Dx Ey F 0 33. 34. y z z x 4y 3z x 7y 8z y 9z z 3 1 5 85 35 3 In Exercises 35–38, use Gaussian elimination to solve the system of equations. 35. 36. 37. 38. y 3y 2x 4x x 3x 4x x x 2y 2x 2x 3y x 3 |
y 3x 3x 2y y 2y 2y 4 4 16 6z z 2z z 13 5z 23 2z 14 z 6 7 3z 11 6z 9 11z 16 7z 11 In Exercises 39 and 40, solve the nonsquare system of equations. 39. 40. 5x 12y 7z 16 3x 7y 4z 9 2x 5y 19z 34 3x 8y 31z 54 In Exercises 41 and 42, find the equation of the parabola y ax 2 bx c that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. 41. y 42. 4 (2, 5) −4 x 4 (1, −2) (0, −5) y 24 12 (−5, 6) −12 −6 (2, 20) 6 (1, 0) x that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. 43. y 1 −5 (2, 1) x 1 432 (5, − 2) (− 1, − 2) 44. y (1, 4) 2 (4, 3) −6 −2 2 4 x (− 2, − 5) −8 45. Data Analysis: Online Shopping The table shows the (in millions) of people shopping (Source: projected numbers online in the United States from 2003 to 2005. eMarketer) y Year Online shoppers, y 2003 2004 2005 101.7 108.4 121.1 (a) Use the technique demonstrated in Exercises 67–70 in Section 7.3 to set up a system of equations for the data and to find a least squares regression parabola that x 3 x models the data. Let corresponding to 2003. represent the year, with (b) Use a graphing utility to graph the parabola and the data in the same viewing window. How well does the model fit the data? (c) Use the model to estimate the number of online shoppers in 2008. Does your answer seem reasonable? 46. Agriculture A mixture of 6 gallons of chemical A, 8 gallons of chemical B, and 13 gallons of chemical C is required to kill a destructive crop insect. Commercial spray X contains 1, 2, and 2 parts, respectively, of these chemicals. Commercial spray Y contains only chemical C. Commercial spray Z contains chemicals A, B, and C in equal amounts |
. How much of each type of commercial spray is needed to get the desired mixture? 47. Investment Analysis An inheritance of $40,000 was divided among three investments yielding $3500 in interest per year. The interest rates for the three investments were 7%, 9%, and 11%. Find the amount placed in each investment if the second and third were $3000 and $5000 less than the first, respectively. 333202_070R.qxd 12/5/05 9:48 AM Page 565 48. Vertical Motion An object moving vertically is at the given heights at the specified times. Find the position equation for the object. s 1 (a) At (b) At seconds, 2 at2 v0t s0 s 134 second, s 86 s 6 s 184 s 116 s 16 seconds, seconds, seconds, second, feet feet feet feet feet feet At At At At 7.4 In Exercises 49–52, write the form of the partial fraction decomposition for the rational expression. Do not solve for the constants. 49. 51. 3 x2 20x 3x 4 x3 5x2 50. 52. x 8 x2 3x 28 x 2 xx2 22 In Exercises 53–60, write the partial fraction decomposition of the rational expression. 53. 55. 57. 59. 4 x x2 6x 8 x2 x2 2x 15 x2 2x x3 x2 x 1 3x2 4x x2 12 54. 56. 58. 60. x x2 3x 2 9 x2 9 4x 3x 12 4x2 x 1x2 1 In Exercises 61–64, sketch the graph of the 7.5 inequality. 61. y ≤ 5 1 2 x 63. y 4x2 > 1 62. 3y x ≥ 7 64. y ≤ 3 x2 2 In Exercises 65–72, sketch the graph and label the vertices of the solution set of the system of inequalities. 65. 67. 3x 2y ≤ 160 y ≤ 180 x ≥ 0 y ≥ 0 x 3x 2y ≥ 24 x 2y ≥ 12 2 ≤ x ≤ 15 y ≤ 15 69. y < x 1 y > x2 1 66. 68. 70. 2x y ≤ 16 x ≥ 0 y ≥ 0 2x 3y ≤ 24 2x y ≥ 16 x 3y ≥ 18 0 ≤ x ≤ 25 0 ≤ y ≤ 25 y ≤ 6 2x x2 y |
≥ x 6 Review Exercises 565 71. 72. 2x 3y ≥ 0 2x y ≤ 8 y ≥ 0 x2 y2 ≤ 9 x 32 y2 ≤ 9 73. Inventory Costs A warehouse operator has 24,000 square feet of floor space in which to store two products. Each unit of product I requires 20 square feet of floor space and costs $12 per day to store. Each unit of product II requires 30 square feet of floor space and costs $8 per day to store. The total storage cost per day cannot exceed $12,400. Find and graph a system of inequalities describing all possible inventory levels. 74. Nutrition A dietitian is asked to design a special dietary supplement using two different foods. Each ounce of food X contains 12 units of calcium, 10 units of iron, and 20 units of vitamin B. Each ounce of food Y contains 15 units of calcium, 20 units of iron, and 12 units of vitamin B. The minimum daily requirements of the diet are 300 units of calcium, 280 units of iron, and 300 units of vitamin B. (a) Write a system of inequalities describing the different amounts of food X and food Y that can be used. (b) Sketch a graph of the region in part (a). (c) Find two solutions to the system and interpret their meanings in the context of the problem. Supply and Demand In Exercises 75 and 76, (a) graph the systems representing the consumer surplus and producer surplus for the supply and demand equations and (b) find the consumer surplus and producer surplus. Demand p 160 0.0001x p 130 0.0002x 75. 76. Supply p 70 0.0002x p 30 0.0003x In Exercises 77– 82, sketch the region determined by 7.6 the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the indicated restraints. 77. Objective function: z 3x 4y Constraints: x ≥ 0 y ≥ 0 2x 5y ≤ 50 4x y ≤ 28 78. Objective function: z 10x 7y Constraints: x ≥ 0 y ≥ 0 2x y ≥ 100 x y ≥ 75 333202_070R.qxd 12/5/05 9:48 AM Page 566 566 Chapter 7 Systems of Equations and Inequalities 79. Objective function: z 1.75x 2.25y Constraints: x ≥ 0 y ≥ 0 2x y |
≥ 25 3x 2y ≥ 45 81. Objective function: z 5x 11y Constraints: x ≥ 0 y ≥ 0 x 3y ≤ 12 3x 2y ≤ 15 80. Objective function: z 50x 70y Constraints: x ≥ 0 y ≥ 0 x 2y ≤ 1500 5x 2y ≤ 3500 82. Objective function: z 2x y Constraints 5x 2y ≥ 20 83. Optimal Revenue A student is working part time as a hairdresser to pay college expenses. The student may work no more than 24 hours per week. Haircuts cost $25 and require an average of 20 minutes, and permanents cost $70 and require an average of 1 hour and 10 minutes. What combination of haircuts and/or permanents will yield an optimal revenue? What is the optimal revenue? 84. Optimal Profit A shoe manufacturer produces a walking shoe and a running shoe yielding profits of $18 and $24, respectively. Each shoe must go through three processes, for which the required times per unit are shown in the table. Process Process Process II III I Hours for walking shoe Hours for running shoe Hours available per day 4 2 24 1 2 9 1 1 8 What is the optimal production level for each type of shoe? What is the optimal profit? 85. Optimal Cost A pet supply company mixes two brands of dry dog food. Brand X costs $15 per bag and contains eight units of nutritional element A, one unit of nutritional element B, and two units of nutritional element C. Brand Y costs $30 per bag and contains two units of nutritional element A, one unit of nutritional element B, and seven units of nutritional element C. Each bag of mixed dog food must contain at least 16 units, 5 units, and 20 units of nutritional elements A, B, and C, respectively. Find the numbers of bags of brands X and Y that should be mixed to produce a mixture meeting the minimum nutritional requirements and having an optimal cost. What is the optimal cost? 86. Optimal Cost Regular unleaded gasoline and premium unleaded gasoline have octane ratings of 87 and 93, respectively. For the week of January 3, 2005, regular unleaded gasoline in Houston, Texas averaged $1.63 per gallon. For the same week, premium unleaded gasoline averaged $1.83 per gallon. Determine the blend of regular and premium unleaded gasoline that results in an optimal cost of midgrade unleaded (89-octane) gasoline. What |
is the optimal (Source: Energy Information Administration) cost? Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 87 and 88, determine whether 87. The system 26 represents the region covered by an isosceles trapezoid. 88. It is possible for an objective function of a linear programming problem to have exactly 10 maximum value points. In Exercises 89–92, find a system of linear equations having the ordered pair as a solution. (There are many correct answers.) 89. 90. 91. 92. 6, 8 5, 4 3, 3 4 1, 9 4 In Exercises 93–96, find a system of linear equations having the ordered triple as a solution. (There are many answers.) 93. 94. 95. 96. 4, 1, 3 3, 5, 6 2, 2 5, 3 4, 2, 8 3 97. Writing Explain what is meant by an inconsistent system of linear equations. 98. How can you tell graphically that a system of linear equa- tions in two variables has no solution? Give an example. 99. Writing Write a brief paragraph describing any advantages of substitution over the graphical method of solving a system of equations. 333202_070R.qxd 12/5/05 9:48 AM Page 567 7 Chapter Test Chapter Test 567 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. In Exercises 1– 3, solve the system by the method of substitution. 1. x y 4x 5y 7 8 2. y x 1 y x 13 3. 2x y2 0 x y 4 In Exercises 4– 6, solve the system graphically. 4. 2x 3y 0 2x 3y 12 5. y 9 x2 y x 3 6. y ln x 7x 2y 11 12 6 In Exercises 7–10, solve the linear system by the method of elimination. 7. 9. 2x 3y 5x 4y x 17 15 2y 3z z 3y z 2x 8. 10. 2.5x y 6 3x 4y 2 3x 2y z 17 x y z 4 x y z 3 11 3 8 In Exercises 11–14, write the partial fraction decomposition of the rational expression. 11. 2x 5 x2 x 2 |
12. 3x2 2x 4 x22 x 13. x2 5 x3 x 14. x2 4 x3 2x In Exercises 15–17, sketch the graph and label the vertices of the solution of the system of inequalities. 15. 2x y ≤ 4 2x y ≥ 0 x ≥ 0 16. y < x2 x 4 y > 4x 17. x2 y2 ≤ x ≥ y ≥ 16 1 3 18. Find the maximum and minimum values of the objective function z 20x 12y and where they occur, subject to the following constraints. x ≥ 0 y ≥ 0 x 4y ≤ 32 3x 2y ≤ 36 Constraints 19. A total of $50,000 is invested in two funds paying 8% and 8.5% simple interest. The yearly interest is $4150. How much is invested at each rate? 20. Find the equation of the parabola y ax 2 bx c passing through the points 0, 6, 2, 2, and 3, 9 2. 21. A manufacturer produces two types of television stands. The amounts (in hours) of time for assembling, staining, and packaging the two models are shown in the table at the left. The total amounts of time available for assembling, staining, and packaging are 4000, 8950, and 2650 hours, respectively. The profits per unit are $30 (model I) and $40 (model II). What is the optimal inventory level for each model? What is the optimal profit? Model Model I 0.5 2.0 0.5 II 0.75 1.5 0.5 Assembling Staining Packaging TABLE FOR 21 333202_070R.qxd 12/5/05 9:48 AM Page 568 Proofs in Mathematics An indirect proof can be useful in proving statements of the form “ implies ” q. p → q q is Recall that the conditional statement is false only when q is false. To prove a conditional statement indirectly, assume that false. If this assumption leads to an impossibility, then you have proved that the conditional statement is true. An indirect proof is also called a proof by contradiction. is true and is true and p p p You can use an indirect proof to prove the following conditional statement, “If a is a positive integer and a2 is divisible by 2, then a is divisible by 2,” as follows. First, assume that aq, is |
true and is odd and can be written as 2. If so, a2 “ is a positive integer and is divisible by 2,” a “ is divisible by 2,” is false. This means that is not divisible by n is an integer. where a 2n 1, ap, a a 2n 1 a2 4n2 4n 1 a2 22n2 2n 1 Definition of an odd integer Square each side. Distributive Property So, by the definition of an odd integer, and you can conclude that a is divisible by 2. a2 is odd. This contradicts the assumption, Example Using an Indirect Proof Use an indirect proof to prove that 2 is an irrational number. Solution Begin by assuming that as the quotient of two integers and 2 a is not an irrational number. Then 2 can be written bb 0 that have no common factors. 2 a b 2 a2 b2 2b2 a2 This implies that 2 is a factor of as is an integer. 2c, c a2. where 2b2 2c2 2b2 4c2 b2 2c2 Assume that 2 is a rational number. Square each side. Multiply each side by b2. So, 2 is also a factor of a, and can be written a Substitute 2c for a. Simplify. Divide each side by 2. and also a factor of So, 2 is a factor of both and This contradicts the assumption that and have no common factors. This implies that 2 is a factor of a So, you can conclude that b is an irrational number. 2 b. a b. b2 568 333202_070R.qxd 12/5/05 9:48 AM Page 569 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. A theorem from geometry states that if a triangle is inscribed in a circle such that one side of the triangle is a diameter of the circle, then the triangle is a right triangle. Show that this theorem is true for the circle x2 y2 100 and the triangle formed by the lines and y 1 y 0, 2 x 5, k2 infinite number of solutions. and k1 2. Find y 2x 20. such that the system of equations has an 3x 5y 8 2x k1y k2 3. Consider the following system of linear equations in and |
x y. ax by e cx dy f Under what conditions will the system have exactly one solution? 4. Graph the lines determined by each system of linear equations. Then use Gaussian elimination to solve each system. At each step of the elimination process, graph the corresponding lines. What do you observe? (a) (b) x 4y 3 5x 6y 13 2x 3y 7 4x 6y 14 5. A system of two equations in two unknowns is solved and has a finite number of solutions. Determine the maximum number of solutions of the system satisfying each condition. (a) Both equations are linear. (b) One equation is linear and the other is quadratic. (c) Both equations are quadratic. 6. In the 2004 presidential election, approximately 118.304 million voters divided their votes among three presidential candidates. George W. Bush received 3,320,000 votes more than John Kerry. Ralph Nader received 0.3% of the votes. Write and solve a system of equations to find the total number of votes cast for each candidate. Let represent the total votes cast for Bush, the total votes cast for Kerry, and N the total votes cast for Nader. (Source: CNN.com) K B 7. The Vietnam Veterans Memorial (or “The Wall”) in Washington, D.C. was designed by Maya Ying Lin when she was a student at Yale University. This monument has two vertical, triangular sections of black granite with a common side (see figure). The bottom of each section is level with the ground. The tops of the two sections can be approximately modeled by the equations 2x 50y 505 and 2x 50y 505 x when the -axis is superimposed at the base of the wall. Each unit in the coordinate system represents 1 foot. How high is the memorial at the point where the two sections meet? How long is each section? −2x + 50y = 505 2x + 50y = 505 Not drawn to scale C2H6 8. Weights of atoms and molecules are measured in atomic mass units (u). A molecule of (ethane) is made up of two carbon atoms and six hydrogen atoms and weighs (propane) is made up of 30.07 u. A molecule of C3H8 three carbon atoms and eight hydrogen atoms and weighs 44.097 u. Find the weights of a carbon atom and a hydrogen atom. 9. To connect a DVD player to a television |
set, a cable with special connectors is required at both ends. You buy a six-foot cable for $15.50 and a three-foot cable for $10.25. Assuming that the cost of a cable is the sum of the cost of the two connectors and the cost of the cable itself, what is the cost of a four-foot cable? Explain your reasoning. 10. A hotel 35 miles from an airport runs a shuttle service to and from the airport. The 9:00 A.M. bus leaves for the airport traveling at 30 miles per hour. The 9:15 A.M. bus leaves for the airport traveling at 40 miles per hour. Write a system of linear equations that represents distance as a function of time for each bus. Graph and solve the system. How far from the airport will the 9:15 A.M. bus catch up to the 9:00 A.M. bus? 569 333202_070R.qxd 12/5/05 9:49 AM Page 570 11. Solve each system of equations by letting X 1x, Y 1y, and Z 1z. 12 x 3 x 12 y 4 y 7 0 (a) (b 13 z 4 10 8 12. What values should be given to a, 1, 2, 3 c and so that the lin- b, as its only solution? ear system shown has x 2y 3z a x y z b 2x 3y 2z c Equation 1 Equation 2 Equation 3 13. The following system has one solution: x 1, y 1, and z 2. 4x 2y 5z 16 x y 0 x 3y 2z 6 Solve the system given by (a) Equation 1 and Equation 2, (b) Equation 1 and Equation 3, and (c) Equation 2 and Equation 3. (d) How many solutions does each of these systems have? 14. Solve the system of linear equations algebraically. x1 3x1 2x1 2x1 x2 2x2 x2 2x2 2x2 2x3 4x3 x3 4x3 4x3 2x4 4x4 x4 5x4 4x4 6x5 12x5 3x5 15x5 13x5 6 14 3 10 13 15. Each day, an average adult moose can process about 32 kilograms of terrestrial vegetation (twigs and leaves) and aquatic vegetation |
. From this food, it needs to obtain about 1.9 grams of sodium and 11,000 calories of energy. Aquatic vegetation has about 0.15 gram of sodium per kilogram and about 193 calories of energy per kilogram, whereas terrestrial vegetation has minimal sodium and about four times more energy than aquatic vegetation. Write and graph a system of inequalities that describes the amounts of terrestrial and aquatic vegetation, respectively, for the daily diet of an average adult moose. (Source: Biology by Numbers) and a t 570 16. For a healthy person who is 4 feet 10 inches tall, the recommended minimum weight is about 91 pounds and increases by about 3.7 pounds for each additional inch of height. The recommended maximum weight is about 119 pounds and increases by about 4.8 pounds for each additional inch of height. (Source: Dietary Guidelines Advisory Committee) (a) Let x be the number of inches by which a person’s height exceeds 4 feet 10 inches and let be the person’s weight in pounds. Write a system of inequalities that describes the possible values of for a healthy person. and y x y (b) Use a graphing utility to graph the system of inequalities from part (a). (c) What is the recommended weight range for someone 6 feet tall? 17. The cholesterol in human blood is necessary, but too much cholesterol can lead to health problems. A blood cholesterol test gives three readings: LDL (“bad”) cholesterol, HDL (“good”) cholesterol, and total cholesterol (LDL HDL). It is recommended that your LDL cholesterol level be less than 130 milligrams per deciliter, your HDL cholesterol level be at least 35 milligrams per deciliter, and your total cholesterol level be no more than 200 milligrams per deciliter. (Source: WebMD, Inc.) (a) Write a system of linear inequalities for the represent HDL recommended cholesterol levels. Let y cholesterol and let represent LDL cholesterol. x (b) Graph the system of inequalities from part (a). Label any vertices of the solution region. (c) Are the following cholesterol levels within recommendations? Explain your reasoning. LDL: 120 milligrams per deciliter HDL: 90 milligrams per deciliter Total: 210 milligrams per deciliter (d) Give an example of cholesterol levels in which the LDL cholesterol level is too high but the HDL and total cholesterol levels are acceptable. (e) Another recommendation is that the ratio |
of total cholesterol to HDL cholesterol be less than 4. Find a point in your solution region from part (b) that meets this recommendation, and explain why it meets the recommendation. 333202_0800.qxd 12/5/05 10:52 AM Page 571 Matrices and Determinants 8.1 Matrices and Systems of Equations 8.2 8.3 8.4 8.5 Operations with Matrices The Inverse of a Square Matrix The Determinant of a Square Matrix Applications of Matrices and Determinants 88 Matrices can be used to analyze financial information such as the profit a fruit farmer makes on two fruit crops AT I O N S Matrices have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Electrical Network, Exercise 82, page 585 • Profit, Exercise 67, page 600 • Long-Distance Plans, Exercise 66, page 634 • Data Analysis: Snowboarders, • Investment Portfolio, Exercise 90, page 585 Exercises 67–70, page 609 • Agriculture, Exercise 61, page 599 • Data Analysis: Supreme Court, Exercise 58, page 630 571 333202_0801.qxd 12/5/05 10:59 AM Page 572 572 Chapter 8 Matrices and Determinants 8.1 Matrices and Systems of Equations What you should learn • Write matrices and identify their orders. • Perform elementary row operations on matrices. • Use matrices and Gaussian elimination to solve systems of linear equations. • Use matrices and Gauss- Jordan elimination to solve systems of linear equations. Why you should learn it You can use matrices to solve systems of linear equations in two or more variables. For instance, in Exercise 90 on page 585, you will use a matrix to find a model for the number of people who participated in snowboarding in the United States from 1997 to 2001. Matrices In this section, you will study a streamlined technique for solving systems of linear equations. This technique involves the use of a rectangular array of real numbers called a matrix. The plural of matrix is matrices. Definition of Matrix n If rectangular array m and are positive integers, an m n n (read “ by ”) matrix is a m Row 1 Row 2 Row 3... Row m a21 a11 a31... am1 Column 1 Column 2 Column 3. a12 a22 a13 a23. Column n a1 |
n a2n............. a3n... amn a32... am2 a33... am3 in which each entry, n rows and columns. Matrices are usually denoted by capital letters. of the matrix is a number. An m n ai j, matrix has m a ij. For instance, The entry in the ith row and jth column is denoted by the double subscript refers to the entry in the second row, third column. notation m n, m n. n A matrix having m rows and columns is said to be of order If a11, a22, a33,... n. the matrix is square of order For a square matrix, the entries are the main diagonal entries. a23 Example 1 Order of Matrices Determine the order of each matrix. a. 2 c. 0 0 0 0 b. d Solution a. This matrix has one row and one column. The order of the matrix is b. This matrix has one row and four columns. The order of the matrix is c. This matrix has two rows and two columns. The order of the matrix is d. This matrix has three rows and two columns. The order of the matrix is 1 1. 1 4. 2 2. 3 2. The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. Now try Exercise 1. A matrix that has only one row is called a row matrix, and a matrix that has only one column is called a column matrix. 333202_0801.qxd 12/5/05 10:59 AM Page 573 Section 8.1 Matrices and Systems of Equations 573 The vertical dots in an augmented matrix separate the coefficients of the linear system from the constant terms. A matrix derived from a system of linear equations (each written in standard form with the constant term on the right) is the augmented matrix of the system. Moreover, the matrix derived from the coefficients of the system (but not including the constant terms) is the coefficient matrix of the system. System: 5 3 6 4y 3y x 2x x 1 1 1 2 1 2 3z z 4z Augmented Matrix: Coefficient Matrix:......... 5 3 6 Note the use of 0 for the missing coefficient of the -variable in the third equation, and also note the fourth column of constant terms in the augmented matrix. |
y When forming either the coefficient matrix or the augmented matrix of a system, you should begin by vertically aligning the variables in the equations and using zeros for the coefficients of the missing variables. Example 2 Writing an Augmented Matrix Write the augmented matrix for the system of linear equations. x 3y w y 4z 2w x 5z 6w 2x 4y 3z 9 2 0 4 What is the order of the augmented matrix? Solution Begin by rewriting the linear system and aligning the variables. x 3y w 9 y 4z 2w 2 x 5z 6w 0 2x 4y 3z 4 R1 R2 R3 R4 1 Next, use the coefficients and constant terms as the matrix entries. Include zeros for the coefficients of the missing variables. 0 4 5 3............ The augmented matrix has four rows and five columns, so it is a The notation is represented by matrix. is used to designate each row in the matrix. For example, Row 1 R1 Rn Now try Exercise 9. 333202_0801.qxd 12/5/05 10:59 AM Page 574 574 Chapter 8 Matrices and Determinants Elementary Row Operations In Section 7.3, you studied three operations that can be used on a system of linear equations to produce an equivalent system. 1. Interchange two equations. 2. Multiply an equation by a nonzero constant. 3. Add a multiple of an equation to another equation. In matrix terminology, these three operations correspond to elementary row operations. An elementary row operation on an augmented matrix of a given system of linear equations produces a new augmented matrix corresponding to a new (but equivalent) system of linear equations. Two matrices are row-equivalent if one can be obtained from the other by a sequence of elementary row operations. Elementary Row Operations 1. Interchange two rows. 2. Multiply a row by a nonzero constant. 3. Add a multiple of a row to another row. Although elementary row operations are simple to perform, they involve a lot of arithmetic. Because it is easy to make a mistake, you should get in the habit of noting the elementary row operations performed in each step so that you can go back and check your work. Example 3 Elementary Row Operations a. Interchange the first and second rows of the original matrix. Original Matrix New Row-Equivalent Matrix R2 R1. Multiply the first row of the original matrix by 1 2. Original Matrix 2 0 |
2 4 3 2 6 3 1 New Row-Equivalent Matrix 1 1 2 2R1 0 3 2 2 →1 3 3 1 1 5 c. Add times the first row of the original matrix to the third row. Original Matrix New Row-Equivalent Matrix 1 0 0 2 3 3 4 2 13 3 1 8 2R1 R3 → Note that the elementary row operation is written beside the row that is changed. Now try Exercise 25. 2 1 5 2 Te c h n o l o g y Most graphing utilities can perform elementary row operations on matrices. Consult the user’s guide for your graphing utility for specific keystrokes. After performing a row operation, the new row-equivalent matrix that is displayed on your graphing utility is stored in the answer variable. You should use the answer variable and not the original matrix for subsequent row operations. 333202_0801.qxd 12/5/05 10:59 AM Page 575 Section 8.1 Matrices and Systems of Equations 575 In Example 3 in Section 7.3, you used Gaussian elimination with backsubstitution to solve a system of linear equations. The next example demonstrates the matrix version of Gaussian elimination. The two methods are essentially the same. The basic difference is that with matrices you do not need to keep writing the variables. Example 4 Comparing Linear Systems and Matrix Operations 9 5 17 R1 R2 Linear System x 2y x 3y 2x 5y 3z 5z 9 4 17 Add the first equation to the second equation. x 2y 3z y 3z 2x 5y 5z 2 Add to the third equation. times the first equation x 2y 3z y 3z y z 9 5 1 Add the second equation to the third equation. x 2y 3z y 3z 2z 9 5 4 Multiply the third equation by 1 2. Associated Augmented Matrix 1 1 2 2 3 5 3 0 5......... 9 4 17 →1 0 2 9 5 17 R1 3 3 5 Add the first row to the. second row 2 1 5 2 Add to the third row 2 1 1 R2......... times the first row. 2R1 R3... 9... 5... 1 3 3 1 1 0 0 2R1 R3 → Add the second row to the. R2 R3 third row. 2.. 3... 1 3... 0 |
2 1 9 5 4 0 0 R2 R3 → Multiply the third row by 1 2 R3 1 2. 2 1 0 3 3 1......... 9 5 2 Remember that you should check a solution by substituting x, y, the values of into each equation of the original system. For example, you can check the solution to Example 4 as follows. and z Equation 1: 1 21 32 9 ✓ Equation 2: 1 31 4 ✓ Equation 3: 21 51 52 17 ✓ x 2y 3z 9 y 3z 5 z 2 1 0 0 1 2R3 → At this point, you can use back-substitution to find x and y. y 32 5 Substitute 2 for z. y 1 Solve for y. x 21 32 9 x 1 y 1, x 1, The solution is Substitute 1 for y and 2 for z. Solve for x. and z 2. Now try Exercise 27. 333202_0801.qxd 12/5/05 10:59 AM Page 576 576 Chapter 8 Matrices and Determinants The last matrix in Example 4 is said to be in row-echelon form. The term echelon refers to the stair-step pattern formed by the nonzero elements of the matrix. To be in this form, a matrix must have the following properties. Row-Echelon Form and Reduced Row-Echelon Form A matrix in row-echelon form has the following properties. 1. Any rows consisting entirely of zeros occur at the bottom of the matrix. 2. For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading 1). 3. For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row. A matrix in row-echelon form is in reduced row-echelon form if every column that has a leading 1 has zeros in every position above and below its leading 1. Example 5 Row-Echelon Form Determine whether each matrix is in row-echelon form. If it is, determine whether the matrix is in reduced row-echelon form.1 a. c. e. d. f Solution The matrices in (a), (c), (d), and (f) are in row-echelon form. The matrices in (d |
) and (f) are in reduced row-echelon form because every column that has a leading 1 has zeros in every position above and below its leading 1. The matrix in (b) is not in row-echelon form because a row of all zeros does not occur at the bottom of the matrix. The matrix in (e) is not in row-echelon form because the first nonzero entry in Row 2 is not a leading 1. Now try Exercise 29. Every matrix is row-equivalent to a matrix in row-echelon form. For instance, in Example 5, you can change the matrix in part (e) to row-echelon form by multiplying its second row by 1 2. 333202_0801.qxd 12/5/05 10:59 AM Page 577 Section 8.1 Matrices and Systems of Equations 577 Gaussian Elimination with Back-Substitution Gaussian elimination with back-substitution works well for solving systems of linear equations by hand or with a computer. For this algorithm, the order in which the elementary row operations are performed is important. You should operate from left to right by columns, using elementary row operations to obtain zeros in all entries directly below the leading 1’s. Example 6 Gaussian Elimination with Back-Substitution Solve the system x 2x x y z 2y z 4y z 4y 7z 2w 3w w 3 2 2 19. Solution 2R1 R1 R3 R4 6R2 R4 1 3R3 13R4 1 R2 0 2 1 1 2 1 0 R1 1 1 1 → 13 0 2 1 1............................................................ 3 2 2 19 2 3 2 19 2 3 6 21 2 3 6 39 2 3 2 3 Write augmented matrix. Interchange R1 and R2 so first column has leading 1 in upper left corner. Perform operations on R3 and R4 so first column has zeros below its leading 1. Perform operations on R4 so second column has zeros below its leading 1. Perform operations on R3 and R4 so third and fourth columns have leading 1’s. The matrix is now in row-echelon form, and the corresponding system is x 2y z y |
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