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point P1 and a terminal point P2, and v has an initial point P3 and a terminal point P4. 52. P1 = (β1, 4), P2 = (3, 1), P3 = (5, 5) and 53. P1 = (6, 11), P2 = (β2, 8), P3 = (0, β 1) and P4 = (9, 2) P4 = (β8, 2) For the following exercises, use the vectors u = 2i β j, v = 4i β 3j, and w = β2i + 5j to evaluate the expre... |
3 ξͺ to rectangular coordinates. 2. Find the area of the triangle in Figure 1. Round each answer to the nearest tenth. 6.25 5 60Β° 7 Figure 1 4. Convert (2, 2) to polar coordinates, and then plot the point. 6. Convert the polar equation to a Cartesian equation: x2 + y2 = 5y. 7. Convert to rectangular form and graph: r =... |
the same direction as v. 26. Given vector v has an initial point P1 = (2, 2) and terminal point P2 = (β1, 0), write the vector v in terms of i and j. On the graph, draw v, and β v. 9 Systems of Equations and Inequalities Figure 1 enigma machines like this one, once owned by Italian dictator Benito Mussolini, were used... |
however, we will study systems of equations on their own: linear and nonlinear, and then partial fractions. We will not be breaking any secret codes here, but we will lay the foundation for future courses. 757 7 58 CHAPTER 9 systems oF eQuations and ineQualities leARnInG OBjeCTIVeS In this section, you will: β’ Solve s... |
the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. Shortly we will investigate methods of finding such a solution if it exists. 2(4) + (7) = 15 True 3(4) β (7) = 5 True In addition to considering the number of eq... |
equations and an ordered pair, determine whether the ordered pair is a solution. Independent System Inconsistent System Dependent System 1. Substitute the ordered pair into each equation in the system. 2. Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a sol... |
β2x β8 β5 β4 β3 β2 (β3, β2) y = x + 1 21 3 4 5 x Figure 4 SECTION 9.1 systems oF linear eQuations: two variaBles 761 The lines appear to intersect at the point (β3, β2). We can check to make sure that this is the solution to the system by substituting the ordered pair into both equations. 2(β3) + (β2) = β8 β8 = β8 Tru... |
we will solve the first equation for y. βx + y = β5 2x β 5y = 1 βx + y = β5 y = x β 5 7 62 CHAPTER 9 systems oF eQuations and ineQualities Now we can substitute the expression x β 5 for y in the second equation. 2x β 5y = 1 2x β 5(x β 5) = 1 2x β 5x + 25 = 1 β3x = β24 x = 8 Now, we substitute x = 8 into the first equa... |
of the original equations and solve for the second variable. 5. Check the solution by substituting the values into the other equation. SECTION 9.1 systems oF linear eQuations: two variaBles 763 Example 4 Solving a System by the Addition Method Solve the given system of equations by addition. x + 2y = β1 βx + y = 3 Sol... |
(11) β3x + 6y = β33 Multiply both sides by β3. Use the distributive property. Now, letβs add them. 3x + 5y = β11 β3x + 6y = β33 11y = β44 y = β4 For the last step, we substitute y = β4 into one of the original equations and solve for x. 3x + 5y = β 11 3x + 5( β 4) = β 11 3x β 20 = β 11 3x = 9 x = 3 Our solution is the ... |
5x β 10y = 30 5(β2) β 10(β4) = 30 β10 + 40 = 30 30 = 30 y 5 4 3 2 1 β1 β1 β2 β3 β4 β5 β6 β7 2x + 3y = β16 β6 β5 β4 β3 β2 (β2, β4) 5x β 10y = 30 21 3 4 5 6 x Example 7 Using the Addition Method in Systems of Equations Containing Fractions Solve the given system of equations in two variables by addition. Figure 7 x y __... |
System of Equations Solve the following system of equations. x = 9 β 2y x + 2y = 13 Solution We can approach this problem in two ways. Because one equation is already solved for x, the most obvious step is to use substitution. x + 2y = 13 (9 β 2y) + 2y = 13 9 + 0y = 13 9 = 13 Clearly, this statement is a contradiction... |
2 3x + 9y = 6 Solution With the addition method, we want to eliminate one of the variables by adding the equations. In this case, letβs focus on eliminating x. If we multiply both sides of the first equation by β3, then we will be able to eliminate the x-variable. 7 68 CHAPTER 9 systems oF eQuations and ineQualities N... |
axis represents quantity in hundreds of units. The y-axis represents either cost or revenue in hundreds of dollars. SECTION 9.1 systems oF linear eQuations: two variaBles 769 ) 70 60 50 40 30 20 10 0 Profit (7, 33) Cost Break-even Revenue β10 0 5 10 Quantity (in hundreds of units) 15 20 Figure 10 The point at which the... |
units is also $77,500. To make a profit, the business must produce and sell more than 50,000 units. See Figure 11. We see from the graph in Figure 12 that the profit function has a negative value until x = 50,000, when the graph crosses the x-axis. Then, the graph emerges into positive y-values and continues on this p... |
this to write an equation for the revenue. 25c + 50a = 70,000 We now have a system of linear equations in two variables. c + a = 2,000 25c + 50a = 70,000 In the first equation, the coefficient of both variables is 1. We can quickly solve the first equation for either c or a. We will solve for a. c + a = 2,000 Substitu... |
exercises, determine whether the given ordered pair is a solution to the system of equations. 6. 5x β y = 4 x + 6y = 2 and (4, 0) 9. β2x + 5y = 7 7. β3x β 5y = 13 8. 3x + 7y = 1 β x + 4y = 10 and (β6, 1) 2x + 4y = 0 and (2, 3) 10. x + 8y = 43 2x + 9y = 7 and (β1, 1) 3x β 2y = β1 and (3, 5) For the following exercises,... |
1 1 __ __ y = β x β 2 8 43 ___ 120 2 1 1 __ __ __ y = x + 28. 9 3 9 1 4 1 __ __ __ y = β x + β 3 5 2 For the following exercises, solve each system by any method. 31. 5x + 9y = 16 x + 2y = 4 32. 6x β 8y = β0.6 3x + 2y = 0.9 33. 5x β 2y = 2.25 7x β 4y = 3 34. x β y = β 5 ___ 12 55 ___ 12 5 __ β6x + y = 2 55 ___ 2 7 72 ... |
0.01x + 0.12y = 0.62 0.15x + 0.20y = 0.52 50. β0.71x + 0.92y = 0.13 0.83x + 0.05y = 2.1 48. 0.5x + 0.3y = 4 0.25x β 0.9y = 0.46 exTenSIOnS For the following exercises, solve each system in terms of A, B, C, D, E, and F where A β F are nonzero numbers. Note that A β B and AE β BD. 51 52. x + Ay = 1 x + By = 1 53. Ax + y... |
If each meal is then sold for $15, after how many meals does the restaurant break even? 62. A number is 9 more than another number. Twice the sum of the two numbers is 10. Find the two numbers. 64. A moving company charges a flat rate of $150, and an additional $5 for each box. If a taxi service would charge $20 for e... |
much was invested in each account? 72. If an investor invests a total of $25,000 into two bonds, one that pays 3% simple interest, and the 7 __ % interest, and the investor other that pays 2 8 earns $737.50 annual interest, how much was invested in each account? 74. CDs cost $5.96 more than DVDs at All Bets Are Off El... |
pattern. We will solve this and similar problems involving three equations and three variables in this section. Doing so uses similar techniques as those used to solve systems of two equations in two variables. However, finding solutions to systems of three equations requires a bit more organization and a touch of vis... |
are those which, after elimination, result in an expression that is always true, such as 0 = 0. Graphically, an infinite number of solutions represents a line or coincident plane that serves as the intersection of three planes in space. β’ Systems that have no solution are those that, after elimination, result in a sta... |
-two system. 4. Back-substitute known variables into any one of the original equations and solve for the missing variable. Example 2 Solving a System of Three Equations in Three Variables by Elimination Find a solution to the following system: x β 2y + 3z = 9 βx + 3y β z = β6 2x β 5y + 5z = 17 (1) (2) (3) Solution Ther... |
invested $4,000 more in mutual funds than he invested in municipal bonds. The total interest earned in one year was $670. How much did he invest in each type of fund? Solution To solve this problem, we use all of the information given and set up three equations. First, we assign a variable to each of the three investm... |
35,000 z = 7,000 y + 4(7,000) = 31,000 y = 3,000 x + 3,000 + 7,000 = 12,000 x = 2,000 John invested $2,000 in a money-market fund, $3,000 in municipal bonds, and $7,000 in mutual funds. Try It #1 Solve the system of equations in three variables. 2x + y β 2z = β1 3x β 3y β z = 5 x β 2y + 3z = 6 Identifying Inconsistent... |
the same location. Therefore, the system is inconsistent. Try It #2 Solve the system of three equations in three variables. x + y + z = 2 y β 3z = 1 2x + y + 5z = 0 expressing the Solution of a System of Dependent equations Containing Three Variables We know from working with systems of equations in two variables that... |
2x y = 2 5 __ y = x 2 3 5 x ξͺ. In this solution, x can be any real number. The values of y and z are dependent So the general solution is ξ’ x, __ __ x, 2 2 on the value selected for x. Analysis As shown in Figure 5, two of the planes are the same and they intersect the third plane on a line. The solution set is infini... |
x + 4y + 5z = β1 and (0, 1, β1) βx + 2y + 3z = β1 8. 6x β 7y + z = 2 βx β y + 3z = 4 and (4, 2, β6) 2x + y β z = 1 10. βx β y + 2z = 3 5x + 8y β 3z = 4 and (4, 1, β7) βx + 3y β 5z = β5 7. 6x β y + 3z = 6 3x + 5y + 2z = 0 and (3, β3, β5) x + y = 0 9 and (4, 4, β1) x β y + z = β1 For the following exercises, solve each ... |
x + 2y + z = 11 βx + 5y + 3z = 4 19. 2x + 3y β 6z = 1 β4x β 6y + 12z = β2 x + 2y + 5z = 10 22. 10x + 2y β 14z = 8 βx β 2y β 4z = β1 β12x β 6y + 6z = β12 24. 5x β 3y + 4z = β1 β4x + 2y β 3z = 0 βx + 5y + 7z = β11 25. x + y + z = 0 2x β y + 3z = 0 x β z = 0 7 82 CHAPTER 9 systems oF eQuations and ineQualities 26. 3x + 2y... |
3 5 __ __ β __ __ __ __ z = y β x β 31. β 4 2 3 4 1 1 1 __ __ __ __ __ __ __ __ y + 1 __ x β 1 __ 34. β 1 __ 8 3 3 6 z = β 23 ___ x β 7 __ y + 1 __ β 2 __ 3 3 3 8 y + 5 __ x β 5 __ β 1 __ z = 0 8 3 6 37. 0.1x β 0.2y + 0.3z = 2 0.5x β 0.1y + 0.4z = 8 0.7x β 0.2y + 0.3z = 8 38. 0.2x + 0.1y β 0.3z = 0.2 0.8x + 0.4y β 1.2... |
4x + 0.2y + 0.1z = 1.6 45. 0.8x + 0.8y + 0.8z = 2.4 0.3x β 0.5y + 0.2z = 0 0.1x + 0.2y + 0.3z = 0.6 exTenSIOnS For the following exercises, solve the system for x, y, and z. 46. x + y + z = 3 47. 5x β 3y β z + 1 _____ 2 = 1 __ 2 + 2z = β3 6x + y β 9 _____ 2 x + 8 _____ 2 β 4y + z = 4 48. x + 4 _____ 7 x β 2 _____ 4 x +... |
for you and your other roommate. The total bill was $82. She forgot to save the individual receipts but remembered that your groceries were $0.05 cheaper than half of her groceries, and that your other roommateβs groceries were $2.10 more than your groceries. How much was each of your share of the groceries? 57. Three... |
of what your third roommateβs supplies were before tax, how much did each of you spend? Give your answer both with and without taxes. 58. At a carnival, $2,914.25 in receipts were taken at the end of the day. The cost of a childβs ticket was $20.50, an adult ticket was $29.75, and a senior citizen ticket was $15.25. T... |
Japan, and China. In millions of barrels per day, the three top countries consumed 39.8% of the worldβs consumed oil. The United States consumed 0.7% more than four times Chinaβs consumption. The United States consumed 5% more than triple Japanβs consumption. What percent of the world oil consumption did the United St... |
6, 2014, http://scaruffi.com/politics/oil.html. 29 βOil reserves, production and consumption in 2001,β accessed April 6, 2014, http://scaruffi.com/politics/oil.html. 30 βOil reserves, production and consumption in 2001,β accessed April 6, 2014, http://scaruffi.com/politics/oil.html. 31 βUSA: The coming global oil cris... |
same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes. Intersection of a Parabola and a Line There are three possible types of soluti... |
equation to solve for x. Always substitute the value into the linear equation to check for extraneous solutions. x β y = β1 x β (2) = β1 x = 1 x β (1) = β1 x = 0 The solutions are (1, 2) and (0, 1), which can be verified by substituting these (x, y) values into both of the original equations. See Figure 31 (1, 2) (0, ... |
One of the equations has already been solved for y. We will substitute y = 3x β 5 into the equation for the circle. Now, we factor and solve for x. x 2 + (3x β 530x + 25 = 5 10 x 2 β30x + 20 = 0 10( x 2 β 3x + 2) = 0 10(x β 2)(x β 1) = 0 x = 2 x = 1 Substitute the two x-values into the original linear equation to solv... |
β’ Two solutions. The circle and the ellipse intersect at two points. β’ Three solutions. The circle and the ellipse intersect at three points. β’ Four solutions. The circle and the ellipse intersect at four points. No solution One solution Two solutions Three solutions Four solutions Figure 6 Example 3 Solving a System ... |
, y > a, or less than, y < a, the graph is drawn with a dashed line. When the inequality is greater than or equal to, y β₯ a, or less than or equal to, y β€ a, the graph is drawn with a solid line. The graphs will create regions in the plane, and we will test each region for a solution. If one point in the region works, ... |
Since y > x 2 + 1 has a greater than symbol, we draw the graph with a dashed line. Then we choose points to test both inside and outside the parabola. Letβs test the points (0, 2) and (2, 0). One point is clearly inside the parabola and the other point is clearly outside0) 2 + 1 2 > 1 True 0 > (2) 2 + 1 0 > 5 False SE... |
(β2 92 CHAPTER 9 systems oF eQuations and ineQualities The two points of intersection are (2, 4) and (β2, 4). Notice that the equations can be rewritten as follows β€ 12 y β€ β 2 x 2 + 12 Graph each inequality. See Figure 10. The feasible region is the region between the two equations bounded by 2 x 2 + y β€ 12 on the to... |
10 15. x 2 + y 2 + = 2500 y = 2 x 2 1 _ 16 For the following exercises, use any method to solve the system of nonlinear equations. 16. β2 x 2 + y = β5 6x β y = 9 19 22 17. β x 2 + y = 2 βx + y = 2 20 23 For the following exercises, use any method to solve the nonlinear system. 24 27 30. x 2 + y 2 = 25 x 2 β y 2 = 36 2... |
variables. 48. = 24 + 1 __ y 2 + 4 = 0 4 __ x 2 β 2 __ y 2 5 __ x 2 49. 6 __ x 2 1 __ x 2 β 1 __ y 2 β 6 __ y 2 = 8 = 1 __ 8 50. x 2 β xy + y 2 β2 = 0 x + 3y = 4 51. x 2 β xy β 2 y 2 β 52. x 2 + 4xy β TeCHnOlOGY For the following exercises, solve the system of inequalities. Use a calculator to graph the system to conf... |
ucible quadratic factor. 9.4 PARTIAl FRACTIOnS Earlier in this chapter, we studied systems of two equations in two variables, systems of three equations in three variables, and nonlinear systems. Here we introduce another way that systems of equations can be utilizedβthe decomposition of rational expressions. Fractions... |
of the linear factors as the denominator. In other words, using the example above, the factors of x 2 β x β 6 are (x β 3)(x + 2), the denominators of the decomposed rational expression. So we will rewrite the simplified form as the sum of individual fractions and use a variable for each numerator. Then, we will solve ... |
B _____ (x β 1) Multiply both sides of the equation by the common denominator to eliminate the fractions: (x + 2)(x β 1) ξ° 3x __________ (x + 2)(x β 1) ξ² = ξ (x + 2) (x β 1) ξ° A ______ ξ (x + 2) ξ² + (x + 2) ξ (x β 1) ξ° B _____ ξ² ξ (x β 1) The resulting equation is Expand the right side of the equation and collect like... |
method, so the decompositions are the same using either method. 3x ___________ (x + 2)(x β 1) = 2 ______ (x + 2) + 1 _____ (x β 1) Although this method is not seen very often in textbooks, we present it here as an alternative that may make some partial fraction decompositions easier. It is known as the Heaviside metho... |
expression with repeated linear factors. β x 2 + 2x + 4 ___________ x 3 β 4 x 2 + 4x Solution The denominator factors are x (x β 2) 2. To allow for the repeated factor of (x β 2), the decomposition will include three denominators: x, (x β 2), and (x β 2) 2. Thus, Next, we multiply both sides by the common denominator.... |
(x β 2) + 2 _______ (x β 2) 2 Try It #2 Find the partial fraction decomposition of the expression with repeated linear factors. 6x β 11 _______ (x β 1) 2 SECTION 9.4 partial Fractions 799 Decomposing Where Q(x ) Has a nonrepeated Irreducible Quadratic Factor P(x ) ____ Q(x ) So far, we have performed partial fraction ... |
+ + equations to solve for the numerators. Example 3 Decomposing When Q(x) Contains a Nonrepeated Irreducible Quadratic Factor P(x) ____ Q(x) Find a partial fraction decomposition of the given expression. 8 x 2 + 12x β 20 ________________ (x + 3)( x 2 + x + 2) Solution We have one linear factor and one irreducible qua... |
+ 3B + Cx + 3C 8 x 2 + 12x β 20 = (2 + B) x 2 + (2 + 3B + C)x + (4 + 3C) Setting the coefficients of terms on the right side equal to the coefficients of terms on the left side gives the system of equations. 2 + B = 8 2 + 3B + C = 12 4 + 3C = β20 (1) (2) (3) Solve for B using equation (1) and solve for C using equatio... |
) has a repeated irreducible quadratic factor P(x) ____ Q(x) P(x) ____ Q(x) of P(x) is less than the degree of Q(x), is The partial fraction decomposition of, when Q(x) has a repeated irreducible quadratic factor and the degree P(x) _____________ n (a x 2 + bx + c) = A 1 x + B 1 ____________ (a x 2 + bx + c) + A 2 x + ... |
1) 2 = A __ x + Bx + C _______ ( x 2 + 1) + Dx + E _______ ( x 2 + 1) 2 We eliminate the denominators by multiplying each term by x ( x 2 + 1) 2. Thus) 2 + (Bx + C)(x)( x 2 + 1) + (Dx + E)(x) Expand the right side. Now we will collect like terms( x 4 + 2 x 2 + 1 + Cx + D x 2 + Ex = A x 4 + 2A + Cx + D x 2 + Ex A + B) ... |
9.4 SeCTIOn exeRCISeS VeRBAl 1. Can any quotient of polynomials be decomposed into at least two partial fractions? If so, explain why, and if not, give an example of such a fraction 2. Can you explain why a partial fraction decomposition is unique? (Hint: Think about it as a system of equations.) 3. Can you explain ho... |
14 ____________ 2 x 2 + 12x + 18 21. x _______ (x β 2) 2 24. β24x β 27 _________ (6x β 7) 2 22. 7x + 14 _______ (x + 3) 2 25. 5 β x _______ (x β 7) 2 27. 5 x 2 + 20x + 8 ___________ 2x (x + 1) 2 28. 4 x 2 + 55x + 25 ____________ 5x (3x + 5) 2 29. 54 x 3 + 127 x 2 + 80x + 16 ____________________ 2 x 2 (3x + 2) 2 30. x ... |
2 x 2 + 14x + 15 _________________ ( x 2 + 4) 2 47. x 2 + 5x + 5 _________ (x + 2) 2 50. 2 x 3 + 11x + 7x + 70 ________________ (2 x 2 + x + 14) 2 53. 2x β 9 _______ ( x 2 β x) 2 exTenSIOnS 45. 48. x 3 + 6 x 2 + 5x + 9 ______________ ( x 2 + 1) 2 x 3 + 2 x 2 + 4x ___________ ( x 2 + 2x + 9) 2 46. 49 ___________ ( x 2 ... |
$300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the eq... |
on. Each entry in a matrix is referred to as a ij, such that i represents the row and j represents the column. Matrices are often referred to by their dimensions: m Γ n indicating m rows and n columns. Example 1 Given matrix A: Finding the Dimensions of the Given Matrix and Locating Entries a. What are the dimensions ... |
row 1, column 1, b 11, of B. Continue the pattern until all entries have been added 10 ξ² 9 Example 4 Finding the Difference of Two Matrices Find the difference of A and B. A = ξ° β2 3 0 1 ξ² and B = ξ° 8 1 5 4 ξ² Solution We subtract the corresponding entries of each matrix. A β B = ξ° β10 2 ξ² β5 β3 808 CHAPTER 9 systems o... |
ar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication. Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in... |
lying the Matrix by a Scalar Multiply matrix A by the scalar 3. Solution Multiply each entry in A by the scalar 3. A = ξ° 8 1 ξ² 5 4 3A = 24 3 ξ² 15 12 = ξ° = ξ° 810 CHAPTER 9 systems oF eQuations and ineQualities Try It #2 Given matrix B, find β2B where A = ξ° 4 1 3 2 ξ² Example 7 Finding the Sum of Scalar Multiples Find the... |
as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers. To obtain the entries in row i of AB, we multiply the entries in row i of A by column j in B and add. For example, given matrices A and B, where the dimensions of A are 2 Γ 3 and the dimensions of B are 3 ... |
13 β
b 32 a 21 β
b 12 + a 22 β
b 22 + a 23 β
b 32 a 11 β
b 13 + a 12 β
b 23 + a 13 β
b 33 ξ² a 21 β
b 13 + a 22 β
b 23 + a 23 β
b 33 properties of matrix multiplication For the matrices A, B, and C the following properties hold. β’ Matrix multiplication is associative: β’ Matrix multiplication is distributive: (AB)C = A(... |
ξ² = ξ° β7 10 ξ² 30 11 b. The dimensions of B are 3 Γ 2 and the dimensions of A are 2 Γ 3. The inner dimensions match so the product is defined and will be a 3 Γ 3 matrix. BA = ξ° = ξ° = ξ° 5 β1 β4 0 2 3 ξ² ξ° β1 2 3 4 0 5 ξ² 5(β1) + β1(4) 5(2) + β1(0) 5(3) + β1(5) β4(β1) + 0(4) β4(2) + 0(0) β4(3) + 0(5) 2(β1) + 3(4) 2(2) + 3(0... |
300(10) + 10(24) + 30(20)] = [2,520 3,840] The total cost for equipment for the Wildcats is $2,520, and the total cost for equipment for the Mud Cats is $3,840. How To⦠Given a matrix operation, evaluate using a calculator. 1. Save each matrix as a matrix variable [A], [B], [C],... 2. Enter the operation into the calc... |
? If so, explain how; if not, explain why. 5. Does matrix multiplication commute? That is, does AB = BA? If so, prove why it does. If not, explain why it does not. AlGeBRAIC 2. Can we multiply any column matrix by any row matrix? Explain why or why not. 4. Can any two matrices of the same size be multiplied? If so, exp... |
10E SECTION 9.5 section exercises 815 For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: A 2 = A β
A) A = ξ° β10 20 5 25 ξ², B = ξ° 40 10 β20 30 ξ², C = ξ° β1 0 0 β1 1 0 ξ² 30. AB 36. C 2 31. BA 37. B 2 A ... |
of linear equations using matrices. 9.6 SOlVInG SYSTeMS WITH GAUSSIAn elIMInATIOn Figure 1 German mathematician Carl Friedrich Gauss (1777β1855). Carl Friedrich Gauss lived during the late 18th century and early 19th century, but he is still considered one of the most prolific mathematicians in history. His contributi... |
, write an augmented matrix. 1. Write the coefficients of the x-terms as the numbers down the first column. 2. Write the coefficients of the y-terms as the numbers down the second column. 3. If there are z-terms, write the coefficients as the numbers down the third column. 4. Draw a vertical line and write the constant... |
row operations on a matrix is the method we use for solving a system of equations. In order to solve the system of equations, we want to convert the matrix to row-echelon form, in which there are ones down the main diagonal from the upper left corner to the lower right corner, and zeros in every position below the mai... |
0 1 ξ² The first step of the Gaussian strategy includes obtaining a 1 as the first entry, so that row 1 may be used to alter the rows below. How Toβ¦ Given an augmented matrix, perform row operations to achieve row-echelon form. 1. The first equation should have a leading coefficient of 1. Interchange rows or multiply b... |
a System of Equations Use Gaussian elimination to solve the given 2 Γ 2 system of equations. 2x + y = 1 4x + 2y = 6 Solution Write the system as an augmented matrix. 2 1 ξ° 4 2 1 6 β£ ξ² Obtain a 1 in row 1, column 1. This can be accomplished by multiplying the first row by 1 __. 2 Next, we want a 0 in row 2, column 1. M... |
β6 0 1 β2 21 __ 0 0 1 2 β£ β£ β£ β£ ξ² ξ² Try It #4 Write the system of equations in row-echelon form. x β 2y + 3z = 9 βx + 3y = β 4 2x β 5y + 5z = 17 Solving a System of linear equations Using Matrices We have seen how to write a system of equations with an augmented matrix, and then how to use row operations and back-subs... |
β1 2x + 3y = 2 y β 2z = 0 Solution Write the augmented matrix. β1 β2 1 2 3 0 0 1 β2 ξ° β1 2 0 ξ² β£ First, multiply row 1 by β1 to get a 1 in row 1, column 1. Then, perform row operations to obtain row-echelon form. β R 1 β ξ° β1 β2 1 2 3 0 0 1 β2 β1 1 0 1 β2 0 2 3 0 2 β1 1 0 1 β2 0 0 β1 2 0 1 β2 1 0 0 0 0 β£ β£ ξ² β£ β£ ξ² The... |
3y + 9z = β1 β2x + 3y β z = β2 βx β 4y + 5z = 1 Solution Write the augmented matrix for the system of equations. On the matrix page of the calculator, enter the augmented matrix above as the matrix variable [A]. 5 3 9 β2 3 β1 β1 β4 5 ξ° β1 ξ² β2 β1 β£ [A] = ξ° 5 3 9 β1 β2 3 β1 β2 β1 β4 5 1 ξ² Use the ref( function in the c... |
twice the amount invested at 5%. How much was invested at each rate? Solution We have a system of three equations in three variables. Let x be the amount invested at 5% interest, let y be the amount invested at 8% interest, and let z be the amount invested at 9% interest. Thus, x + y + z = 10,000 0.05x + 0.08y + 0.09z... |
Solve a System of Two equations Using an Augmented Matrix (http://openstaxcollege.org/l/system2augmat) β’ Solve a System of Three equations Using an Augmented Matrix (http://openstaxcollege.org/l/system3augmat) β’ Augmented Matrices on the Calculator (http://openstaxcollege.org/l/augmatcalc) 826 CHAPTER 9 systems oF eQu... |
the system by Gaussian elimination. 13. ξ° 3 2 0 3 ξ² β1 β9 4 β1 8 8 5 7 β£ 1 0 16. ξ° 0 0 20. ξ° β2 0 0 2 1 ξ² β1 β£ 3 ξ² 0 β£ 1 0 17. ξ° 1 0 1 ξ² 2 β£ 21. 2x β 3y = β 9 5x + 4y = 58 1 2 18. ξ° 4 5 3 ξ² 6 β£ 22. 6x + 2y = β4 3x + 4y = β17 β1 2 19. ξ° 4 β5 β3 6 β£ ξ² 23. 2x + 3y = 12 4x + y = 14 24. β4x β 3y = β2 3x β 5y = β13 25. β5x ... |
y + 1 __ 3 z = β 53 ___ 14 z = 3 z = 23 ___ 15 SECTION 9.6 section exercises 827 38. x + y β 4z = β4 5x β 3y β 2z = 0 2x + 6y + 7z = 30 41. x + 2y β z = 1 βx β 2y + 2z = β2 3x + 6y β 3z = 3 x β 2 __ 44. 1 __ z = β 1 __ 4 3 2 x + 1 __ y = 4 __ 1 __ 7 5 3 z = 2 __ y β 1 __ 1 __ 5 9 3 37. β2x + 3y β 2z = 3 4x + 2y β z = ... |
2z = β1 z + 5 _____ ReAl-WORlD APPlICATIOnS For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. 52. Every day, a cupcake store sells 5,000 cupcakes in 53. At a competing cupcake store, $4,520 worth of chocolate and vanilla flavors. If the chocolate... |
up 12% of total ice cream sales. This year, the same three ice creams made up 16.9% of ice cream sales. The rocky road sales doubled, the banana sales increased by 50%, and the pumpkin sales increased by 20%. If the rocky road ice cream had one less percent of sales than the banana ice cream, find out the percentage o... |
1 __ 2 and ξ’ 1 __ ξͺ 2 = 1. The multiplicative inverse of a matrix is similar in concept, except that the product of matrix A and its 2 inverse A β1 equals the identity matrix. The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. We identify identity matrices by I n w... |
1. Given matrix A of order n Γ n and matrix B of order n Γ n multiply AB. 2. If AB = I, then find the product BA. If BA = I, then B = A β1 and A = B β1. Example 2 Showing That Matrix A Is the Multiplicative Inverse of Matrix B Show that the given matrices are multiplicative inverses of each other. 1 5 β2 β9 β9 β5 2 1 ... |
β2 ξ² 2 β3 ξ² Find the product of the two matrices on the left side of the equal sign. ξ² = ξ° 1 β2 ξ² ξ° 2 β ξ² = ξ° Next, set up a system of equations with the entry in row 1, column 1 of the new matrix equal to the first entry of the identity, 1. Set the entry in row 2, column 1 of the new matrix equal to the corresponding... |
β2 and add to row 2. 4. Add row 2 to row 1. 5. Multiply row 2 by β1. The matrix we have found is A β1 β2 1 ξ² 5 β2 1 0 ξ° 0 β1 3 β1 5 β2 ξ² 1 0 ξ° 0 1 3 β1 ξ² β5 2 β£ A β1 = ξ° 3 β1 ξ² β5 2 Finding the Multiplicative Inverse of 2 Γ 2 Matrices Using a Formula When we need to find the multiplicative inverse of a 2 Γ 2 matrix, w... |
. A β1 = ξ° 1 β1 2 3 ξ² Example 5 Finding the Inverse of the Matrix, If It Exists Find the inverse, if it exists, of the given matrix. A = ξ° 3 6 1 2 ξ² Solution We will use the method of augmenting with the identity. Switch row 1 and row 2. 2. Multiply row 1 by β3 and add it to row 2. There is nothing further we can do. T... |
β1 A = I. A β1 = B = ξ° β1 1 0 β1 0 1 6 β2 β1 1 0 β1 0 1 6 β2 β1 1 0 β1 0 1 6 β2 β1 = ξ° = ξ° = ξ° A β(β1) + 3(β1) + 1(6) 2(1) + 3(0) + 1(β2) 2(0) + 3(1) + 1(β3) 3(β1) + 3(β1) + 1(6) 3(1) + 3(0) + 1(β2) 3(0) + 3(1) + 1(β3) 2(β1) + 4(β1) + 1(6) 2(1) + 4(0) + 1(β2) 2(0) + 4(1) + 1(β3) ξ² β1(2) + 1(3) + 0(2) β1(2) + 0(3) + 1(... |
a system AX = B. For example, look at the following system of equations. From this system, the coefficient matrix is The variable matrix is And the constant matrix is Then AX = B looks like Recall the discussion earlier in this section regarding multiplying a real number by its inverse, ( 2 β1 ) 2 = ξ’ 1 __ ξͺ 2 = 1. To... |
Using the formula to calculate the inverse of a 2 by 2 matrix, we have: ξ² 8 3 ξ° 4 11 1 = d βb 1 ξ² ξ° _______ ad β bc βc a 11 β8 ξ² β4 3 1 ξ° ___________ 3(11) β 8(4) = = 1 __ ξ° 1 11 β8 ξ² β4 3 So, 11 β8 ξ² β4 3 Now we are ready to solve. Multiply both sides of the equation by A β1. A β1 = ξ° ( A β1 )AX = ( A β1 )B 11 β8 ξ² ξ°... |
56 β4 β11 β41 β1 β3 β11 ξ² ξ° 35 ξ² β26 β7 1 1 _ 5 56 __ 5 0 3 0 1 0 β4 β11 β41 β3 β1 0 0 β11 0 1 x ξ² = ξ° ξ° y z β£ β£ β£ ξ² ξ² 56 __ 5 1 _ 5 0 0 19 __ 5 4 _ 5 1 0 β11 1 0 0 1 0 β1 3 1 β3 3 1 ξ° 0 1 0 0 56 __ 5 1 _ 5 0 0 19 __ __ 5 19 __ 5 1 _ 5 β 11 _ 5 4 _ 5 1 _ 5 β __ 5 19 __ 5 1 β 11 _ 5 4 _ 5 1 β3 0 1 0 5 0 ξ² ξ² β£ β£ β£ ξ° ξ° ξ°... |
x β 17y + 11z = 0 βx + 11y β 7z = 8 3y β 2z = β2 How Toβ¦ Given a system of equations, solve with matrix inverses using a calculator. 1. Save the coefficient matrix and the constant matrix as matrix variables [A] and [B]. 2. Enter the multiplication into the calculator, calling up each matrix variable as needed. 3. If t... |
4. Can a matrix with an entire column of zeros have an entire row of zeros can have an inverse? inverse? Explain why or why not. 5. Can a matrix with zeros on the diagonal have an inverse? If so, find an example. If not, prove why not. For simplicity, assume a 2 Γ 2 matrix. AlGeBRAIC In the following exercises, show t... |
y = 9 12x + 4y = β6 32. β2x + 3y = 3 ___ 10 β x + 5y = 1 __ 2 33. 8 __ y = 2 __ x β 4 __ 5 5 5 y = 7 ___ β 8 __ x + 1 __ 5 5 10 34. 1 __ 2 1 __ 2 y = β 1 __ x + 1 __ 5 4 y = β 9 __ x β 3 __ 5 4 For the following exercises, solve a system using the inverse of a 3 Γ 3 matrix. 35. 3x β 2y + 5z = 21 5x + 4y = 37 x β 2y β 5... |
β 43 ___ 44. β 1 __ 20 2 2 y = 31 ___ 5 __ x + 11 ___ 5 4 2 46. 0.5x β 3y + 6z = β0.8 0.7x β 2y = β0.06 0.5x + 4y + 5z = 0 For the following exercises, find the inverse of the given matrix. 47. 51 ξ² ξ° 48. ξ° β1 0 2 β3 1 0 ξ² 492 3 β5 0 1 1 ξ² ξ° 50 ξ² ξ° 842 CHAPTER 9 systems oF eQuations and ineQualities ReAl-WORlD APPlICA... |
foot, the wood $10 per square foot, and the plywood $5 per square foot. The farmer spent a total of $51, and the total amount of materials used was 14 ft 2. He used 3 ft 2 more chicken wire than plywood. How much of each material in did the farmer use? 57. A clothing store needs to order new inventory. It has three di... |
can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a square matrix to determine whether there is a solution to the system of equations. Perhaps on... |
method, such as elimination, will have to be used. To understand Cramerβs Rule, letβs look closely at how we solve systems of linear equations using basic row operations. Consider a system of two equations in two variables1) (2) We eliminate one variable using row operations and solve for the other. Say that we wish t... |
β 126 ___ 42 = β3 15 3 13 β3 _ = 12 3 2 β3 β£ β£ 12 15 2 13 _ 12 3 2 β3 β£ β£ β£ β£ β£ β£ Solve for y. The solution is (2, β3). Try It #1 Use Cramerβs Rule to solve the 2 Γ 2 system of equations. x + 2y = β11 β2x + y = β13 evaluating the Determinant of a 3 Γ 3 Matrix Finding the determinant of a 2 Γ 2 matrix is straightforwar... |
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