6.7 Solving Systems of Linear Equations 363 a) Write the cost and revenue equations. C x 15 400 = + R x 25 = b) Graph both equations, for up to and including 50 backpacks, on the same axes. * c) Use the graph to estimate the number of backpacks Benjamin’s Backpacks must sell to break even. 40 backpacks d) Write the profit formula. P x 10 400 = − e) Use the profit formula to determine whether Benjamin’s Backpacks makes a profit or loss if it sells 30 backpacks. What is the profit or loss? Loss of $100 f) How many backpacks must Benjamin’s Backpacks sell to make a profit of $1000? 140 backpacks 52. MODELING—Manufacturing Car Bicycle Racks A manufacturer sells bicycle racks that mount on the backs of cars for $150 per unit. Manufacturing costs consist of a fixed cost of $3850 and a production cost of $115 per unit. a) Write the cost and revenue equations. C x R x 115 3850, 150 = + = b) Graph both equations for up to and including 150 units on the same axes. * c) Use the graph to estimate the number of units the manufacturer must sell to break even. 110 units d) Write the profit formula. P x 35 3850 = − e) Use the profit formula to determine the manufacturer’s profit or loss if 100 units are sold. Loss of $350 f) How many units must the manufacturer sell to make a profit of $875? 135 units 53. MODELING—Selling Earbuds SoundsGreat manufactures and sells earbuds. They sell their earbuds for $40 per unit. The manufacturing costs consist of a fixed cost of $4050 and a production cost of $15 per unit. a) Write the cost and revenue equations. C x R x 15 4050, 40 = + = b) Graph both equations for up to and including 200 units on the same axes. * c) Use the graph to estimate the number of units SoundsGreat must sell to break even. 162 units d) Write the profit formula. P x 25 4050 = − e) Use the profit formula to determine whether SoundsGreat makes a profit or loss if 155 units are sold. Loss of $175 f) How many units must SoundsGreat sell in order to make a profit of $575? 185 units *See Instructor Answer Appendix $ $ 45. x y x y 2 7 1 4 14 3 − = − = No solution; inconsistent system 46. x y x y 4 6 8 2 13 + = − − = No solution; inconsistent system 47. x y x y 3 2 8 5 3 2 + = − = 28 19 , 34 19 ⎛ ⎝⎜ ⎞ ⎠⎟ 48. x y x y 6 6 1 4 9 4 + = + = 1 2 , 2 3 ⎛ − ⎝⎜ ⎞ ⎠⎟ Problem Solving In Exercises 49 –54, part of the question involves determining a system of equations that models the situation. 49. MODELING—Truck Rentals The cost of renting a mediumsized truck at Ugly Truck Rental is $30 per day plus $0.79 a mile. The cost of renting a similar truck at Discount Rentals is $24 per day plus $0.85 a mile. a) Write a system of equations, with one equation representing the total cost of renting a truck from Ugly Truck for a day and the other equation representing the total cost of renting a truck from Discount Rentals for a day. * b) Graph both equations for up to and including 200 miles on the same axes. * c) Use the graph to estimate the number of miles that would need to be driven in a day for the cost of renting a truck from Ugly Truck to equal the cost of renting a truck from Discount Rentals. 100 miles 50. MODELING—Landscaping Costs Tom’s Tree and Landscape Service charges $200 for a consultation fee plus $60 per hour for labor, and Lawn Perfect Landscape Service charges $305 for a consultation fee plus $25 per hour for labor. a) Write the system of equations to represent the cost of the two landscaping services. * b) Graph both equations for up to and including 10 hours on the same axes. * c) Use the graph to estimate the number of hours of landscaping that must be used for both services to have the same cost. 3 hours 51. MODELING—Selling Backpacks Benjamin’s Backpacks can sell backpacks for $25 per backpack. The costs for making the backpacks are a fixed cost of $400 and a production cost of $15 per backpack. $ $ $ Karamysh/Shutterstock
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