810 CHAPTER 11 Systems of Equations and Inequalities In Problems 65–70 show that each matrix has no inverse. 65. 4 2 2 1 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 66. 3 1 2 6 1 − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 67. 15 3 10 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 68. ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 3 0 4 0 69. 3 1 1 1 4 7 1 2 5 − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 70. 1 1 3 2 4 1 5 7 1 − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 87. College Tuition Nikki and Joe take classes at a community college, LCCC, and a local university, SIUE. The number of credit hours taken and the cost per credit hour (2024–2025 academic year, tuition and approximate fees) are as follows: Applications and Extensions (a) Write a matrix A for the credit hours taken by each student and a matrix B for the cost per credit hour. (b) Compute AB and interpret the results. Sources: lc.edu, siue.edu (a) Write a matrix A for the amounts borrowed by each student and a matrix B for the monthly interest rates. (b) Compute AB and interpret the result. (c) Let C 1 1 . = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Compute A C B ( ) + and interpret the result. LCCC SIUE Nikki 6 9 Joe 3 12 Cost per Credit Hour LCCC $186.00 SIUE $516.85 Lender 1 Lender 2 Jamal $4000 $3000 Stephanie $2500 $3800 Monthly Interest Rate Lender 1 0.011 (1.1%) Lender 2 0.006 (0.6%) 88. School Loan Interest Jamal and Stephanie both have school loans issued from the same two banks. The amounts borrowed and the monthly interest rates are given next (interest is compounded monthly). In Problems 71–74, use a graphing utility to find the inverse, if it exists, of each matrix. Round answers to two decimal places. 71. 25 61 12 18 12 7 3 4 1 − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 72. 18 3 4 6 20 14 10 25 15 − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 73. 44 21 18 6 2 10 15 5 21 12 12 4 8 16 4 9 − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 74. 16 22 3 5 21 17 4 8 2 8 27 20 5 15 3 10 − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ In Problems 75–78, use the inverse matrix found in Problem 71 to solve the following systems of equations. Round answers to two decimal places. 75. x y z x y y x y z 25 61 12 10 18 12 7 9 3 4 12 + − = − + =− + − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ 76. x y z x y z x y z 25 61 12 15 18 12 7 3 3 4 12 + − = − + =− + − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ 77. x y z x y z x y z 25 61 12 21 18 12 7 7 3 4 2 + − = − + = + − =− ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ 78. x y z x y z x y z 25 61 12 25 18 12 7 10 3 4 4 + − = − + = + − =− ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ Mixed Practice In Problems 79–86, solve each system of equations using any method you wish. 79. x y x y 2 3 11 5 7 24 + = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ 80. x y x y 2 8 8 7 13 + = − + =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ 81. − + = − + − = − =− ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y 2 4 2 3 5 2 17 4 3 22 82. + − =− + = − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x z y z 2 3 2 4 3 6 6 2 2 83. − + = − + − = + − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y z 5 4 2 5 4 3 7 13 4 17 84. + − = + + =− + − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y z 3 2 2 2 6 7 2 2 14 17 85. − + = − + − =− − + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z y z 2 3 4 3 2 3 5 6 86. − + + = + − =− + + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y z 4 3 2 6 3 2 9 6
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