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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 ... |
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 wi... |
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 m... |
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 s... |
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.... |
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? i... |
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 ... |
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... |
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 ... |
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... |
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 poin... |
) (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 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 ... |
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... |
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 ... |
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... |
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 u... |
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 solvin... |
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 equatio... |
(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 mul... |
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 (ar... |
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 state... |
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 str... |
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 o... |
(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 ... |
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.Edu... |
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 wheth... |
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 calor... |
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% b... |
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,... |
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 wi... |
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 an... |
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 Dra... |
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 follow... |
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 3... |
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 in... |
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 w... |
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 thr... |
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... |
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... |
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... |
. 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 2... |
, 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 poi... |
) 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 corp... |
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 co... |
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)... |
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... |
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,... |
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.... |
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... |
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 var... |
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 fo... |
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 ... |
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 f... |
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 t... |
. 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, w... |
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-po... |
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 syst... |
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 line... |
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, sci... |
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 ... |
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 ... |
(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 pro... |
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 f... |
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.... |
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 sys... |
, 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 l... |
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. (So... |
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 in... |
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 graphi... |
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 v... |
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_070... |
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... |
, 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. 3332... |
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... |
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 ≤ Constrai... |
: 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 C... |
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... |
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 Assemblin... |
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 ... |
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.... |
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 algebraica... |
. 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 Exerc... |
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 1... |
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... |
. 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... |
≥ 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... |
≥ 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 exp... |
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 pr... |
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 t... |
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 d... |
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 1... |
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 hot... |
. 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... |
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 wi... |
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 sec... |
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 ... |
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... |
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... |
) 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 firs... |
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