Chapter Test 851 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? 58. Mixed Nuts A store that specializes in selling nuts has available 72 pounds (lb) of cashews and 120 lb of peanuts. These are to be mixed in 12-ounce (oz) packages as follows: a lower-priced package containing 8 oz of peanuts and 4 oz of cashews, and a quality package containing 6 oz of peanuts and 6 oz 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 numbers of each kind of package. (b) Graph the system and label the corner points. 59. Determining the Speed of the Current of the Aguarico River On a recent trip to the Cuyabeno Wildlife Reserve in the Amazon region of Ecuador, Mike took a 100-kilometer trip by speedboat down the Aguarico River from Chiritza to the Flotel Orellana.As Mike watched the Amazon unfold, he wondered how fast the speedboat was going and how fast the current of the white-water Aguarico River was. Mike timed the trip downstream at 2.5 hours and the return trip at 3 hours. What were the two speeds? 60. 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 would it take each of them, working alone, to finish the job? 61. 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 or 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 are being produced in excess of the number that the factory is obligated to deliver? 62. Describe four ways of solving a system of three linear equations containing three variables. Which method do you prefer? Why? The Chapter Test Prep Videos include step-by-step solutions to all chapter test exercises. These videos are available in MyLab™ Math. In Problems 1–4, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, state that it is inconsistent. In Problems 11 and 12, find the inverse of each nonsingular matrix. 11. = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ A 3 2 5 4 12. = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ B 1 1 1 2 5 1 2 3 0 In Problems 13–16, solve each system of equations using matrices. If the system has no solution, state that it is inconsistent. 13. + = − =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 6 3 12 2 2 14. + = + = ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ x y x y 1 4 7 8 2 56 15. + + =− + + =− + + =− ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y z 2 4 3 2 7 15 12 4 7 13 10 16. + − = − + = + − =− ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y z 2 2 3 5 2 8 3 5 8 2 In Problems 17 and 18, find the value of each determinant. 17. −2 5 3 7 18. − − − 2 4 6 1 4 0 1 2 4 In Problems 19 and 20, use Cramer’s Rule, if possible, to solve each system. 19. + = − − = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 4 3 23 3 5 19 20. − + = − + − =− − + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y z 4 3 2 15 2 3 15 5 5 2 18 Chapter Test 1. − + =− + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 2 7 4 3 9 2. − = − = ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ x y x y 1 3 2 1 5 30 18 3. − + = + − =− + + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y z 2 5 3 4 2 5 2 3 8 4. + − =− − − + = − + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪ x y z x y z x y z 3 2 8 3 2 3 1 6 3 15 8 5. Write the augmented matrix corresponding to the system of equations: − + = − − + =− + − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y x y z 4 5 0 2 6 19 5 5 10 6. Write the system of equations corresponding to the augmented matrix: − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 3 2 4 1 0 8 2 1 3 6 2 11 In Problems 7–10, use the matrices below to compute each expression. = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ A B C 1 1 0 4 3 2 1 2 5 0 3 1 4 6 1 3 1 8 7. +A C 2 8. −A C3 9. CB 10. BA
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