Chapter Review
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Chapter Review Things to Know Systems of equations (pp. 726–728) Systems with no solutions are inconsistent.
Systems with a solution are consistent.
Consistent systems of linear equations have either a unique solution or an infinite number of solutions. Determinants and Cramer’s Rule (pp. 758, 760 and 764) Matrix (pp. 742 and 769)
Rectangular array of numbers, called entries
m by n matrix (p. 769)
Matrix with m rows and n columns
Identity matrix I (p. 777)
Square matrix whose diagonal entries are 1’s, while all other entries are 0’s
Inverse of a matrix (p. 778)
A-1 is the inverse of A if AA-1 = A-1 A = I
Nonsingular matrix (p. 778)
A square matrix that has an inverse
Linear programming (p. 816) Maximize (or minimize) a linear objective function, z = Ax + By, subject to certain conditions, or constraints, expressible as linear inequalities in x and y. A feasible point 1x, y2 is a point that satisfies the constraints of a linear programming problem. Location of solution (p. 818) If a linear programming problem has a solution, it is located at a corner point of the graph of the feasible points. If a linear programming problem has multiple solutions, at least one of them is located at a corner point of the graph of the feasible points. In either case, the corresponding value of the objective function is unique.
Objectives Section 10.1
1 ✓ 2 ✓ 3 ✓ 4 ✓
✓ 6 ✓ 7 ✓ 5
10.2
10.3
✓ 2 ✓ 3 ✓ 4 ✓ 1 ✓ 2 ✓ 1
3 ✓ 4 ✓
✓ 1 ✓ 2 ✓ 3 ✓ 4 ✓ 5 ✓ 5
10.4
You should be able to
Á
Solve systems of equations by substitution (p. 729) Solve systems of equations by elimination (p. 730) Identify inconsistent systems of equations containing two variables (p. 732) Express the solution of a system of dependent equations containing two variables (p. 732) Solve systems of three equations containing three variables (p. 733) Identify inconsistent systems of equations containing three variables (p. 735) Express the solution of a system of dependent equations containing three variables (p. 736) Write the augmented matrix of a system of linear equations (p. 742) Write the system from the augmented matrix (p. 743) Perform row operations on a matrix (p. 744) Solve a system of linear equations using matrices (p. 745) Evaluate 2 by 2 determinants (p. 758) Use Cramer’s Rule to solve a system of two equations containing two variables (p. 759) Evaluate 3 by 3 determinants (p. 762) Use Cramer’s Rule to solve a system of three equations containing three variables (p. 764) Know properties of determinants (p. 765) Find the sum and difference of two matrices (p. 770) Find scalar multiples of a matrix (p. 772) Find the product of two matrices (p. 773) Find the inverse of a matrix (p. 778) Solve a system of linear equations using inverse matrices (p. 781)
Review Exercises 1–14, 101, 102, 105–107 1–14, 101, 102, 105–107 9, 10, 13, 98 14, 97 15–18, 99, 100, 103 18 17 35–44 19, 20 35–44 35–44 45, 46 51–54 47–50 55, 56 57, 58 21, 22 23, 24 25–28 29–34 35–37, 39, 40, 43, 44
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CHAPTER 10
10.5
1 ✓ 2 ✓ 3 ✓ 4 ✓ 1 ✓ 2 ✓ 1 ✓ 2 ✓ 3 ✓ 1 ✓ 2 ✓
10.6 10.7
10.8
Systems of Equations and Inequalities
P , where Q has only nonrepeated linear factors (p. 786) Q P Decompose , where Q has repeated linear factors (p. 788) Q P Decompose , where Q has a nonrepeated irreducible quadratic factor (p. 790) Q P Decompose , where Q has repeated irreducible quadratic factors (p. 791) Q Solve a system of nonlinear equations using substitution (p. 793) Solve a system of nonlinear equations using elimination (p. 795) Graph an inequality by hand (p. 804) Graph an inequality using a graphing utility (p. 806) Graph a system of inequalities (p. 808) Set up a linear programming problem (p. 816) Solve a linear programming problem (p. 816) Decompose
59, 60 61, 62 63, 64, 67, 68 65, 66 69–78 69–78 79–82 79–82 83–92,104 108,109 93–96,108,109
Review Exercises In Problems 1–18, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent. Verify your result using a graphing utility. 3x - 4y = 4 2x + y = 0 2x - y = 5 2x + 3y = 2 1. b 2. b 3. c 4. c 1 13 5x + 2y = 8 7x - y = 3 x - 3y = 5x - 4y = 2 2 6. b
x - 3y + 5 = 0 2x + 3y - 5 = 0
7. b
x - 3y + 4 = 0 3 4 9. c 1 x - y + = 0 2 2 3
10. c
1 y = 2 4 y + 4x + 2 = 0
11. b
3x - 2y = 8 2 x - y = 12 3
14. b
2x + 5y = 10 4x + 10y = 20
x + 2y - z = 6 15. c 2x - y + 3z = -13 3x - 2y + 3z = -16
2x - 4y + z = -15 17. c x + 2y - 4z = 27 5x - 6y - 2z = -3
x - 4y + 3z = 15 18. c -3x + y - 5z = -5 -7x - 5y - 9z = 10
5. b
13. c
x - 2y - 4 = 0 3x + 2y - 4 = 0
x + 5y - z = 2 16. c 2x + y + z = 7 x - y + 2z = 11
x +
y = 2x - 5 x = 3y + 4
8. b
2x + 3y - 13 = 0 3x - 2y = 0
12. b
x = 5y + 2 y = 5x + 2 4x + 5y = 21 5x + 6y = 42
In Problems 19 and 20, write the system of equations corresponding to the given augmented matrix. 19. B
3 1
2 4
`
1 20. C 5 2
8 R -1
2 0 -1
5 -3 0
3
-2 8S 0
In Problems 21–28, use the following matrices to compute each expression. Verify your result using a graphing utility. 1 A = C 2 -1 21. A + C 25. AB
22. A - C 26. BA
0 4 S, 2
B = B
4 1
-3 1
0 R, -2 23. 6A 27. CB
3 C = C1 5
-4 5S 2 24. -4B 28. BC
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Chapter Review
825
In Problems 29–34, find the inverse, if there is one, of each matrix algebraically. If there is not an inverse, say that the matrix is singular. Verify your result using a graphing utility. 1 3 3 4 6 -3 2 29. B 30. B 31. C 1 2 1S R R 1 3 1 -2 1 -1 2 3 1 2 4 -8 -3 1 32. C 3 2 -1 S 33. B 34. B R R -1 2 -6 2 1 1 1 In Problems 35–44, solve each system of equations algebraically using matrices. If the system has no solution, say that it is inconsistent. Verify your result using a graphing utility.
35. b
3x + 2y =
3x - 2y = 1 10x + 10y = 5
6
36. d x - y = -
1 2
2x + y + z = 5 38. c 4x - y - 3z = 1 8x + y - z = 5
39. c
x - y + z = 0 41. c x - y - 5z - 6 = 0 2x - 2y + z - 1 = 0
4x - 3y + 5z = 0 42. c 2x + 4y - 3z = 0 6x + 2y + z = 0
x - 3y + 3z x + 2y 44. d x + 3z + x + y +
t z 2t 5z
5x - 6y - 3z = 6 37. c 4x - 7y - 2z = -3 3x + y - 7z = 1
x - 2z = 1 2x + 3y = -3 4x - 3y - 4z = 3
x + 2y - z = 2 40. c 2x - 2y + z = -1 6x + 4y + 3z = 5 x 2x 43. d x 3x
+ -
y y 2y 4y
+
z z 2z z
+ +
t 2t 3t 5t
= 4 = -3 = 3 = 6
In Problems 45–50, find the value of each determinant algebraically. Verify your result using a graphing utility. 45. `
3 1
2 48. 3 0 -1
4 ` 3
46. ` 3 1 2
10 53 3
0 ` 3
-4 1
2 49. 3 5 2
1 0 6
-3 13 0
1 47. 3 -1 4
4 2 1
0 63 3
-2 50. 3 1 -1
1 2 4
0 33 2
In Problems 51–56, use Cramer’s Rule, if applicable, to solve each system. 51. b
x - 2y = 4 3x + 2y = 4
52. b
x - 3y = -5 2x + 3y = 5
54. b
3x - 4y - 12 = 0 5x + 2y + 6 = 0
x + 2y - z = 6 55. c 2x - y + 3z = -13 3x - 2y + 3z = -16
53. b
2x + 3y - 13 = 0 3x - 2y = 0
x - y + z = 8 56. c 2x + 3y - z = -2 3x - y - 9z = 9
In Problems 57 and 58, use properties of determinants to find the value of each determinant if it is known that
` 57. `
2x y ` 2a b
x y ` =8 a b 58. `
y x ` b a
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= 1 = 3 = 0 = -3
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CHAPTER 10
Systems of Equations and Inequalities
In Problems 59–68, write the partial fraction decomposition of each rational expression. 59. 64.
6 x1x - 42
60.
3x
1x - 221x + 12 2
65.
x 1x + 221x - 32
61.
x3
66.
1x + 42 2
2
x - 4
x 1x - 12
62.
1x + 162
67.
2
x3 + 1
2
2
2x - 6
x
1x - 22 1x - 12
63.
1x + 921x + 12
1x + 121x - 12
68.
1x + 421x2 - 12
2
x2
2
2
2
4
2
In Problems 69–78, solve each system of equations algebraically. Verify your result using a graphing utility. 69. b
2x + y + 3 = 0 x2 + y2 = 5
70. b
x2 + y2 = 16 2x - y2 = -8
71. b
2xy + y2 = 10 3y2 - xy = 2
72. b
3x2 - y2 = 1 7x - 2y2 - 5 = 0
73. b
x2 + y2 = 6y x2 = 3y
74. b
2x2 + y2 = 9 x2 + y2 = 9
75. b
3x2 + 4xy + 5y2 = 8 x2 + 3xy + 2y2 = 0
76. b
3x2 + 2xy - 2y2 = 6 xy - 2y2 + 4 = 0
77.
x2 - 3x + y2 + y = -2 c x2 - x + y + 1 = 0 y
78.
2
x2 + x + y2 = y + 2 c 2 - y x + 1 = x
In Problems 79–82 graph each inequality (a) by hand and (b) by using a graphing utility. 79. 3x + 4y … 12
81. y … x2
80. 2x - 3y Ú 6
82. x Ú y2
In Problems 83–88, graph each system of inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points. 83. b
-2x + y … 2 x + y Ú 2
x y 86. d 3x + y 2x + y
Ú Ú Ú Ú
84. b
x - 2y … 6 2x + y Ú 2
x y 87. d 2x + y x + 2y
0 0 6 2
Ú Ú … Ú
0 0 8 2
x y 85. d x + y 2x + 3y
Ú Ú … …
0 0 4 6
x y 88. d 3x + y 2x + 3y
Ú Ú … Ú
0 0 9 6
In Problems 89–92, graph each system of inequalities. 89. b
x2 + y2 … 16 x + y Ú 2
90. b
y2 … x - 1 x - y … 3
91. b
y … x2 xy … 4
92. b
x2 + y2 Ú 1 x2 + y2 … 4
In Problems 93–96, solve each linear programming problem. 93. Maximize
z = 3x + 4y
subject to
x Ú 0, y Ú 0, 3x + 2y Ú 6, x + y … 8
94. Maximize
z = 2x + 4y
subject to
x Ú 0, y Ú 0, x + y … 6, x Ú 2
95. Minimize
z = 3x + 5y
subject to
x Ú 0, y Ú 0, x + y Ú 1, 3x + 2y … 12, x + 3y … 12
96. Minimize
z = 3x + y
subject to
x Ú 0, y Ú 0, x … 8, y … 6, 2x + y Ú 4
97. Find A so that the system of equations has infinitely many solutions.
b
2x + 5y = 5 4x + 10y = A
98. Find A so that the system in Problem 97 is inconsistent.
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Chapter Review
99. Curve Fitting Find the quadratic function y = ax2 + bx + c that passes through the three points 10, 12, 11, 02, and 1-2, 12. 100. Curve Fitting Find the general equation of the circle that passes through the three points 10, 12, 11, 02, and 1-2, 12.
[Hint: The general equation of a circle is x2 + y2 + Dx + Ey + F = 0.] 101. Blending Coffee A coffee distributor is blending a new coffee that will cost $3.90 per pound. It will consist of a blend of $3.00 per pound coffee and $6.00 per pound coffee. What amounts of each type of coffee should be mixed to achieve the desired blend? [Hint: Assume that the weight of the blended coffee is 100 pounds.]
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in the Amazon region of Ecuador, a 100-kilometer trip by speedboat was taken down the Aguarico River from Chiritza to the Flotel Orellana. As I watched the Amazon unfold, I wondered how fast the speedboat was going and how fast the current of the white-water Aguarico River was. I timed the trip downstream at 2.5 hours and the return trip at 3 hours. What were the two speeds? 106. Finding the Speed of the Jet Stream On a flight between Midway Airport in Chicago and Ft. Lauderdale, Florida, a Boeing 737 jet maintains an airspeed of 475 miles per hour. If the trip from Chicago to Ft. Lauderdale takes 2 hours, 30 minutes and the return flight takes 2 hours, 50 minutes, what is the speed of the jet stream? (Assume that the speed of the jet stream remains constant at the various altitudes of the plane and that the plane flies with the jet stream one way and against it the other way.) 107. Constant Rate Jobs If Bruce and Bryce work together for 1 hour and 20 minutes, they will finish a certain job. If Bryce and Marty work together for 1 hour and 36 minutes, the same job can be finished. If Marty and Bruce work together, they can complete this job in 2 hours and 40 minutes. How long will it take each of them working alone to finish the job?
$3.00/lb
$3.90/lb
$6.00/lb
102. Farming A 1000-acre farm in Illinois is used to grow corn and soy beans. The cost per acre for raising corn is $65, and the cost per acre for soy beans is $45. If $54,325 has been budgeted for costs and all the acreage is to be used, how many acres should be allocated for each crop? 103. Cookie Orders A cookie company makes three kinds of cookies, oatmeal raisin, chocolate chip, and shortbread, packaged in small, medium, and large boxes. The small box contains 1 dozen oatmeal raisin and 1 dozen chocolate chip; the medium box has 2 dozen oatmeal raisin, 1 dozen chocolate chip, and 1 dozen shortbread; the large box contains 2 dozen oatmeal raisin, 2 dozen chocolate chip, and 3 dozen shortbread. If you require exactly 15 dozen oatmeal raisin, 10 dozen chocolate chip, and 11 dozen shortbread, how many of each size box should you buy? 104. Mixed Nuts A store that specializes in selling nuts has available 72 pounds of cashews and 120 pounds of peanuts. These are to be mixed in 12-ounce packages as follows: a lower-priced package containing 8 ounces of peanuts and 4 ounces of cashews and a quality package containing 6 ounces of peanuts and 6 ounces of cashews. (a) Use x to denote the number of lower-priced packages and use y to denote the number of quality packages. Write a system of linear inequalities that describes the possible number of each kind of package. (b) Graph the system and label the corner points. 105. Determining the Speed of the Current of the Aguarico River On a recent trip to the Cuyabeno Wildlife Reserve
108. Maximizing Profit on Figurines A factory manufactures two kinds of ceramic figurines: a dancing girl and a mermaid. Each requires three processes: molding, painting, and glazing. The daily labor available for molding is no more than 90 work-hours, labor available for painting does not exceed 120 work-hours, and labor available for glazing is no more than 60 work-hours. The dancing girl requires 3 workhours for molding, 6 work-hours for painting, and 2 workhours for glazing. The mermaid requires 3 work-hours for molding, 4 work-hours for painting, and 3 work-hours for glazing. If the profit on each figurine is $25 for dancing girls and $30 for mermaids, how many of each should be produced each day to maximize profit? If management decides to produce the number of each figurine that maximizes profit, determine which of these processes has work-hours assigned to it that are not used. 109. Minimizing Production Cost A factory produces gasoline engines and diesel engines. Each week the factory is obligated to deliver at least 20 gasoline engines and at least 15 diesel engines. Due to physical limitations, however, the factory cannot make more than 60 gasoline engines nor more than 40 diesel engines in any given week. Finally, to prevent layoffs, a total of at least 50 engines must be produced. If gasoline engines cost $450 each to produce and diesel engines cost $550 each to produce, how many of each should be produced per week to minimize the cost? What is the excess capacity of the factory; that is, how many of each kind of engine is being produced in excess of the number that the factory is obligated to deliver? 110. Describe four ways of solving a system of three linear equations containing three variables. Which method do you prefer? Why?
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