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Lesson 6.1 Skills Practice Name Date Prepping for the Robot Challenge Solving Linear Systems Graphically and Algebraically Vocabulary Match each term to its corresponding definition. 1. a process of solving a system of equations by substituting a variable in one equation with an equivalent expression a. system of linear equations c. substitution method 2. systems with no solutions b. break-even point e. inconsistent systems 3. the point when the cost and the income are equal c. substitution method b. break-even point 4. systems with one or many solutions d. consistent systems d. consistent systems 5. t wo or more linear equations that define a relationship between quantities e. inconsistent systems © 2012 Carnegie Learning a. system of linear equations 6 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 421 421 4/23/12 12:05 PM Lesson 6.1 Skills Practice page 2 Problem Set Write a system of linear equations to represent each problem situation. Define each variable. Then, graph the system of equations and estimate the break-even point. Explain what the break-even point represents with respect to the given problem situation. 1. Eric sells model cars from a booth at a local flea market. He purchases each model car from a distributor for $12, and the flea market charges him a booth fee of $50. Eric sells each model car for $20. Eric’s income can be modeled by the equation y 5 20x, where y represents the income (in dollars) and x represents the number of model cars he sells. Eric’s expenses can be modeled by the equation y 5 12x 1 50, where y represents the expenses (in dollars) and x represents the number of model cars he purchases from the distributor. y 5 20x y 5 12x 1 50 200 Income y 180 Expenses 160 Dollars 140 120 100 80 60 40 20 0 1 2 3 4 5 6 7 8 Number of Model Cars x 9 10 © 2012 Carnegie Learning The break-even point is between 6 and 7 model cars. Eric must sell more than 6 model cars to make a profit. 6 422 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 422 4/23/12 12:05 PM Lesson 6.1 Skills Practice page 3 Name Date 2. Ramona sets up a lemonade stand in front of her house. Each cup of lemonade costs Ramona $0.30 to make, and she spends $6 on the advertising signs she puts up around her neighborhood. She sells each cup of lemonade for $1.50. Ramona’s income can be modeled by the equation y 5 1.50x, where y represents the income (in dollars) and x represents the number of cups of lemonade she sells. Ramona’s expenses can be modeled by the equation y 5 0.30x 1 6, where y represents the expenses (in dollars) and x represents the number of cups of lemonade she makes. y 5 1.50x y 5 0.30x 1 6 15 Income y Dollars 12 Expenses 9 6 3 0 1 2 3 4 5 6 7 8 Cups of Lemonade x 9 10 © 2012 Carnegie Learning The break-even point is 5 cups of lemonade. Ramona must sell more than 5 cups of lemonade to make a profit. 6 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 423 423 4/23/12 12:05 PM Lesson 6.1 Skills Practice page 4 3. Chen starts his own lawn mowing business. He initially spends $180 on a new lawnmower. For each yard he mows, he receives $20 and spends $4 on gas. Chen’s income can be modeled by the equation y 5 20x, where y represents the income (in dollars) and x represents the number of yards he mows. Chen’s expenses can be modeled by the equation y 5 4x 1 180, where y represents the expenses (in dollars) and x represents the number of yards he mows. y 5 20x y 5 4x 1 180 400 Income y 360 320 Dollars 280 Expenses 240 200 160 120 80 40 0 2 x 4 6 8 10 12 14 16 18 20 Number of Yards Mowed © 2012 Carnegie Learning The break-even point is between 11 and 12 yards mowed. Chen must mow more than 11 yards to make a profit. 6 424 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 424 4/23/12 12:05 PM Lesson 6.1 Skills Practice page 5 Name Date 4. Olivia is building birdhouses to raise money for a trip to Hawaii. She spends a total of $30 on the tools needed to build the houses. The material to build each birdhouse costs $3.25. Olivia sells each birdhouse for $10. Olivia’s income can be modeled by the equation y 5 10x, where y represents the income (in dollars) and x represents the number of birdhouses she sells. Olivia’s expenses can be modeled by the equation y 5 3.25x 1 30, where y represents the expenses (in dollars) and x represents the number of birdhouses she builds. y 5 10x y 5 3.25x 1 30 100 Income y 90 80 Dollars 70 Expenses 60 50 40 30 20 10 0 1 2 3 4 5 6 7 8 Number of Birdhouses x 9 10 © 2012 Carnegie Learning The break-even point is between 4 and 5 birdhouses. Olivia must sell more than 4 birdhouses to make a profit. 6 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 425 425 4/23/12 12:05 PM Lesson 6.1 Skills Practice page 6 5. The Spanish Club is selling boxes of fruit as a fundraiser. The fruit company charges the Spanish Club $7.50 for each box of fruit and a shipping and handling fee of $100 for the entire order. The Spanish Club sells each box of fruit for $15. The Spanish Club’s income can be modeled by the equation y 5 15x, where y represents the income (in dollars) and x represents the number of fruit boxes sold. The Spanish Club’s expenses can be modeled by the equation y 5 7.50x 1 100, where y represents the expenses (in dollars) and x represents the number of fruit boxes ordered. y 5 15x y 5 7.50x 1 100 y 300 270 Income Expenses 240 Dollars 210 180 150 120 90 60 30 0 2 x 4 6 8 10 12 14 16 18 20 Number of Fruit Boxes © 2012 Carnegie Learning The break-even point is between 13 and 14 boxes of fruit. The Spanish Club must sell more than 13 boxes of fruit to make a profit. 6 426 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 426 4/23/12 12:05 PM Lesson 6.1 Skills Practice page 7 Name Date 6. Jerome sells flowers for $12 per bouquet through his Internet flower site. Each bouquet costs him $5.70 to make. Jerome also paid a one-time fee of $150 for an Internet marketing firm to advertise his company. Jerome’s income can be modeled by the equation y 5 12x, where y represents the income (in dollars) and x represents the number of bouquets he sells. Jerome’s expenses can be modeled by the equation y 5 5.70x 1 150, where y represents the expenses (in dollars) and x represents the number of bouquets he makes. y 5 12x y 5 5.70x 1 150 400 y 360 Income 320 Expenses Dollars 280 240 200 160 120 80 40 0 3 x 6 9 12 15 18 21 24 27 30 Number of Bouquets © 2012 Carnegie Learning The break-even point is between 23 and 24 bouquets. Jerome must sell more than 23 bouquets to make a profit. 6 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 427 427 4/23/12 12:05 PM Lesson 6.1 Skills Practice page 8 Transform both equations in each system of equations so that each coefficient is an integer. 1 2 __ x 1 __ 1 y 5 5 3 2 8. 1 3 __ __ x 2 y 5 10 4 4 __ 3 y 5 4 1 x 1 __ 2 2 7. 1 2 __ __ x 2 y 5 7 3 3 __ __ ( __ __ __ __ ( __ __ __ __ ( __ __ __ __ ( __ __ 1 2 x 1 1 y 5 5 2 x 2 1 y 5 7 1 x 1 3 y 5 4 3 x 2 1 y 5 10 3 2 3 3 2 2 4 4 1 3 2 1 3 1 1 2 x 1 y 3 x 2 y 5 7 6 2 x 1 1 y 5 4 10 5 5 4 x 2 y 5 3 3 3 4 4 2 2 2 3x 2 y 5 40 x 1 3y 5 8 2x 2 y 5 21 22x 1 3y 5 30 ) ) __ 5 x 2 3 5 __ 1 y 6 9. 4 1 2 __ __ x 1 y 5 __ 9 5 5 5 __ ( __ __ __ ) ) ) 0.5x 1 1.2y 5 2 10. 3.3x 2 0.7y 5 3 __ __ __ ( __ __ __ ) 9 1 2 x 1 y 5 5 x 2 3 5 1 y 5 5 5 4 6 0.5x 1 1.2y 5 2 3.3x 2 0.7y 5 3 1 5 1 9 2 12 x 2 3 5 10( 0.5x 1 1.2y 5 2 ) 10( 3.3x 2 0.7y 5 3 ) y 5 x 1 y 5 5 4 6 5 5 5x 1 12y 5 20 33x 2 7y 5 30 15x 2 36 5 2y 2x 1 y 5 9 20.1x 2 0.5y 5 1.1 0.2x 2 0.4y 5 2 ) 0.5y 5 1.1 10( 0.2x 2 0.4y 5 2 ) 10 ( 20.1x 2 2x 2 5y 5 11 2x 2 4y 5 20 6 428 0.3y 5 2 2 0.8x 12. 1.1x 5 3y 2 0.4 1.1x 5 3y 2 0.4 0.3y 5 2 2 0.8x 10 2 0.4 ) 10( 0.3y 5 2 2 0.8x ( 1.1x 5 3y ) 3y 5 20 2 8x 11x 5 30y 2 4 © 2012 Carnegie Learning 0.2x 2 0.4y 5 2 11. 20.1x 2 0.5y 5 1.1 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 428 4/23/12 12:05 PM Lesson 6.1 Skills Practice page 9 Name Date Solve each system of equations by substitution. Determine whether the system is consistent or inconsistent. 2x 1 y 5 9 14. y 5 5x 1 2 y 5 2x 2 3 13. x54 y 5 2(4) 2 3 2x 1 (5x 1 2) 5 9 y 5 8 2 3 7x 1 2 5 9 y 5 5 7x 5 7 The solution is (4, 5). x 5 1 The system is consistent. y 5 5(1) 1 2 y 5 5 1 2 y 5 7 The solution is (1, 7). The system is consistent. __ 1 x 1 __ 3 y 5 27 y 5 3x 2 2 15. y 2 3x 5 4 y 2 3x 5 4 y 5 3x 1 4 2 16. 2 1 __ y 5 2x 210 3 3 1 3 1 y 5 2x 210 2 x 1 y 5 27 2 2 3 x 1 3y 5 214 y 5 6x 2 30 ( __ __ ( __ ) ) x 1 3(6x 2 30) 5 214 3x 1 4 5 3x 2 2 x 1 18x 2 90 5 214 4 5 22 © 2012 Carnegie Learning 2 90 5 214 There 19x is no solution. 19x 5 76 The system is inconsistent. x 5 4 6 y 5 6(4) 2 30 y 5 24 2 30 y 5 26 The solution is (4, 26). The system is consistent. Chapter 6 Skills Practice 8069_Skills_Ch06.indd 429 429 4/23/12 12:05 PM Lesson 6.1 Skills Practice page 10 0.8x 2 0.2y 5 1.5 17. 0.1x 1 1.2y 5 0.8 0.8x 2 0.2y 5 1.5 0.1x 1 1.2y 5 0.8 10 ) 10 ) 1.5 0.8 ( 0.8x 2 0.2y 5 ( 0.1x 1 1.2y 5 8x 2 2y 5 15 x 1 12y 5 8 x 5 8 2 12y 8(8 2 12y) 2 2y 5 15 64 2 96y 2 2y 5 15 64 2 98y 5 15 64 5 98y 1 15 x 1 12(0.5) 5 8 49 5 98y x 1 6 5 8 0.5 5 y x 5 2 The solution is (2, 0.5). The system is consistent. 0.3y 5 0.6x 1 0.3 18. 1.2x 1 0.6 5 0.6y 0.3y 5 0.6x 1 0.3 1.2x 1 0.6 5 0.6y 10(0.3y 5 0.6x 1 0.3) 10(1.2x 1 0.6 5 0.6y) 3y 5 6x 1 3 12x 1 6 5 6y y 5 2x 1 1 12x 1 6 5 12x 1 6 050 6 The system has an infinite number of solutions. The system is consistent. 430 © 2012 Carnegie Learning 12x 1 6 5 6(2x 1 1) Chapter 6 Skills Practice 8069_Skills_Ch06.indd 430 4/23/12 12:05 PM Lesson 6.2 Skills Practice Name Date There’s Another Way? Using Linear Combinations to Solve a Linear System Vocabulary Define the term in your own words. 1. linear combinations method The linear combinations method is a process used to solve a system of equations by adding two equations together, resulting in an equation with one variable. Problem Set Write a system of equations to represent each problem situation. Solve the system of equations using the linear combinations method. 1. The high school marching band is selling fruit baskets as a fundraiser. They sell a large basket containing 10 apples and 15 oranges for $20. They sell a small basket containing 5 apples and 6 oranges for $8.50. How much is the marching band charging for each apple and each orange? Let x represent the amount charged for each apple. Let y represent the amount charged for each orange. 10x 1 15y 5 20 10x 1 15y 5 20 5x 1 6y 5 8.50 22(5x 1 6y 5 8.50) 10x 1 15y 5 20 210x 2 12y 5 217 © 2012 Carnegie Learning 3y 5 3 y51 10x 1 15(1) 5 20 6 10x 1 15 5 20 10x 5 5 x 5 0.5 The solution is (0.5, 1). The band charges $0.50 for each apple and $1.00 for each orange. Chapter 6 Skills Practice 8069_Skills_Ch06.indd 431 431 4/23/12 12:05 PM Lesson 6.2 Skills Practice page 2 2. Asna works on a shipping dock at a tire manufacturing plant. She loads a pallet with 4 Mudslinger tires and 6 Roadripper tires. The tires on the pallet weigh 212 pounds. She loads a second pallet with 7 Mudslinger tires and 2 Roadripper tires. The tires on the second pallet weigh 184 pounds. How much does each Mudslinger tire and each Roadripper tire weigh? Let x represent the weight of a Mudslinger tire. Let y represent the weight of a Roadripper tire. 4x 1 6y 5 212 4x 1 6y 5 212 7x 1 2y 5 184 23(7x 1 2y 5 184) 4x 1 6y 5 212 221x 2 6y 5 2552 217x 5 2340 x 5 20 4(20) 1 6y 5 212 80 1 6y 5 212 6y 5 132 y 5 22 The solution is (20, 22). Each Mudslinger tire weighs 20 pounds and each Roadripper tire weighs 22 pounds. 3. The Pizza Barn sells one customer 3 large pepperoni pizzas and 2 orders of breadsticks for $30. They sell another customer 4 large pepperoni pizzas and 3 orders of breadsticks for $41. How much does the Pizza Barn charge for each pepperoni pizza and each order of breadsticks? Let x represent the charge for each pepperoni pizza. Let y represent the charge for each order of breadsticks. 3x 1 2y 5 30 3(3x 1 2y 5 30) 4x 1 3y 5 41 22(4x 1 3y 5 41) 28x 2 6y 5 282 x58 6 3(8) 1 2y 5 30 24 1 2y 5 30 2y 5 6 © 2012 Carnegie Learning 9x 1 6y 5 90 y53 The solution is (8, 3). The Pizza Barn sells each pepperoni pizza for $8 and each order of breadsticks for $3. 432 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 432 4/23/12 12:05 PM Lesson 6.2 Skills Practice page 3 Name Date 4. Nancy and Warren are making large pots of chicken noodle soup. Nancy opens 4 large cans and 6 small cans of soup and pours them into her pot. Her pot contains 115 ounces of soup. Warren opens 3 large cans and 5 small cans of soup. His pot contains 91 ounces of soup. How many ounces of soup does each large can and each small can contain? Let x represent the number of ounces in a large can of soup. Let y represent the number of ounces in a small can of soup. 4x 1 6y 5 115 3(4x 1 6y 5 115) 3x 1 5y 5 91 24(3x 1 5y 5 91) 12x 1 18y 5 345 3x 1 5(9.5) 5 91 212x 2 20y 5 2364 22y 5 219 y 5 9.5 3x 1 47.5 5 91 3x 5 43.5 x 5 14.5 The solution is (14.5, 9.5). Each large can contains 14.5 ounces of soup and each small can contains 9.5 ounces of soup. 5. Taylor and Natsumi are making block towers out of large and small blocks. They are stacking the blocks on top of each other in a single column. Taylor uses 4 large blocks and 2 small blocks to make a tower 63.8 inches tall. Natsumi uses 9 large blocks and 4 small blocks to make a tower 139.8 inches tall. How tall is each large block and each small block? © 2012 Carnegie Learning Let x represent the height (in inches) of each large block. Let y represent the height (in inches) of each small block. 4x 1 2y 5 63.8 22(4x 1 2y 5 63.8) 9x 1 4y 5 139.8 9x 1 4y 5 139.8) 28x 2 4y 5 2127.6 4(12.2) 1 2y 5 63.8 9x 1 4y 5 139.8 x 5 12.2 48.8 1 2y 5 63.8 6 2y 5 15 y 5 7.5 The solution is (12.2, 7.5). Each large block is 12.2 inches tall and each small block is 7.5 inches tall. Chapter 6 Skills Practice 8069_Skills_Ch06.indd 433 433 4/23/12 12:05 PM Lesson 6.2 Skills Practice page 4 6. Dave has 2 buckets that he uses to fill the water troughs on his horse farm. He wants to determine how many ounces each bucket holds. On Tuesday, he fills an empty 2000 ounce water trough with 7 large buckets and 5 small buckets of water. On Thursday, he fills the same empty water trough with 4 large buckets and 10 small buckets of water. How many ounces does each bucket hold? Let x represent the number of ounces the large bucket holds. Let y represent the number of ounces the small bucket holds. 7x 1 5y 5 2000 22(7x 1 5y 5 2000) 4x 1 10y 5 2000 4x 1 10y 5 2000 214x 2 10y 5 24000 4x 1 10y 5 2000 210x 5 22000 x 5 200 7(200) 1 5y 5 2000 1400 1 5y 5 2000 5y 5 600 y 5 120 The solution is (200, 120). The large bucket holds 200 ounces. The small bucket holds 120 ounces. Solve each system of equations using the linear combinations method. 3x 1 5y 5 8 7. 2x 2 5y 5 22 4x 2 y 5 2 8. 2x 1 2y 5 26 3x 1 5y 5 8 2(4x 2 y 5 2) 2x 2 5y 5 22 2x 1 2y 5 26 5x 5 30 x56 8x 2 2y 5 4 10x 5 30 18 1 5y 5 8 x53 5y 5 210 y 5 22 The solution is (6, 22). 6 2(3) 1 2y 5 26 6 1 2y 5 26 2y 5 26 y 5 10 The solution is (3, 10). 434 © 2012 Carnegie Learning 2x 1 2y 5 26 3(6) 1 5y 5 8 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 434 4/23/12 12:05 PM Lesson 6.2 Skills Practice page 5 Name Date 10x 2 6y 5 26 9. 5x 2 5y 5 5 2x 2 4y 5 4 10. 23x 1 10y 5 14 10x 2 6y 5 26 22(5x 2 5y 5 5) 10x 2 6y 5 26 3(2x 2 4y 5 4) 2(23x 1 10y 5 14) 6x 2 12y 5 12 210x 1 10y 5 210 26x 1 20y 5 28 4y 5 216 8y 5 40 y 5 24 y55 5x 2 5(24) 5 5 2x 2 4(5) 5 4 5x 1 20 5 5 2x 2 20 5 4 5x 5 215 2x 5 24 x 5 23 x 5 12 The solution is (23, 24). 3x 1 2y 5 14 11. 4x 1 5y 5 35 The solution is (12, 5). x 1 6y 5 11 12. 2x 2 12y 5 10 © 2012 Carnegie Learning 5(3x 1 2y 5 14) 2(x 1 6y 5 11) 22(4x 1 5y 5 35) 2x 2 12y 5 10 15x 1 10y 5 70 2x 1 12y 5 22 28x 2 10y 5 270 2x 2 12y 5 10 7x 5 0 4x 5 32 x50 x58 3(0) 1 2y 5 14 0 1 2y 5 14 2y 5 14 y57 6 8 1 6y 5 11 6y 5 3 y 5 0.5 The solution is (8, 0.5). The solution is (0, 7). Chapter 6 Skills Practice 8069_Skills_Ch06.indd 435 435 4/23/12 12:05 PM Lesson 6.2 Skills Practice 1.5x 1 1.2y 5 0.6 13. 0.8x 2 0.2y 5 2 page 6 3 1 3 x 1 __ __ y 5 2 __ 2 4 14. 4 2 2 2 __ __ __ x 1 y 5 3 3 3 10(1.5x 1 1.2y 5 0.6) 3 3 1 4 x 1 y 5 2 4 2 4 10(0.8x 2 0.2y 5 2) 2 2 3 2 x 1 y 5 3 3 3 15x 1 12y 5 6 ( __ __ __ ) ( __ __ __ ) 8x 2 2y 5 20 3x 1 2y 5 23 2x 1 2y 5 2 15x 1 12y 5 6 6(8x 2 2y 5 20) 3x 1 2y 5 23 21(2x 1 2y 5 2) 15x 1 12y 5 6 48x 2 12y 5 120 3x 1 2y 5 23 63x 5 126 22x 2 2y 5 22 x52 x 5 25 15(2) 1 12y 5 6 2(25) 1 2y 5 2 30 1 12y 5 6 210 1 2y 5 2 12y 5 224 2y 5 12 y 5 22 y56 The solution is (25, 6). © 2012 Carnegie Learning The solution is (2, 22). 6 436 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 436 4/23/12 12:05 PM Lesson 6.3 Skills Practice Name Date What’s For Lunch? Solving More Systems Problem Set Write a system of equations to represent each problem situation. Solve the system of equations using any method. Then, answer any associated questions. 1. Jason and Jerry are competing at a weightlifting competition. They are both lifting barbells containing 200 pounds of plates (weights). Jason’s barbell has 4 large and 10 small plates on it. Jerry’s barbell has 6 large and 5 small plates on it. How much does each large plate and each small plate weigh? Let x represent the weight (in pounds) of a large plate. Let y represent the weight (in pounds) of a small plate. 4x 1 10y 5 200 6x 1 5y 5 200 One possible solution path: Linear Combinations Method: 4x 1 10y 5 200 22(6x 1 5y 5 200) 4x 1 10y 5 200 212x 2 10y 5 2400 28x 5 2200 © 2012 Carnegie Learning x 5 25 4(25) 1 10y 5 200 100 1 10y 5 200 10y 5 100 y 5 10 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 437 6 The solution is (25, 10). Each large plate weighs 25 pounds. Each small plate weighs 10 pounds. 437 4/23/12 12:05 PM Lesson 6.3 Skills Practice page 2 2. Rachel needs to print some of her digital photos. She is trying to choose between Lightning Fast Foto and Snappy Shots. Lightning Fast Foto charges a base fee of $5 plus an additional $0.20 per photo. Snappy Shots charges a base fee of $7 plus an additional $0.10 per photo. Determine the number of photos for which both stores will charge the same amount. Explain which store Rachel should choose depending on the number of photos she needs to print. Let x represent the number of photos printed. Let y represent the total cost (in dollars) to print x photos. y 5 0.20x 1 5 Lightning Fast Foto y 5 0.10x 1 7 Snappy Shots One possible solution path: Graphing Method: 20 y Printing Cost (dollars) 18 16 Lightning Fast Foto y 5 0.20x 1 5 14 12 Snappy Shots y 5 0.10x 1 7 10 8 6 The solution is (20, 9). Both stores charge $9.00 for printing 20 photos. If Rachel wants to print fewer than 20 photos, then she should choose Lightning Fast Foto. If Rachel wants to print more than 20 photos, then she should choose Snappy Shots. 4 2 4 x 8 12 16 20 24 28 32 36 40 Number of Photos Printed © 2012 Carnegie Learning 0 6 438 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 438 4/23/12 12:05 PM Lesson 6.3 Skills Practice Name page 3 Date 3. Raja is trying to decide which ice cream shop is the better buy. Cold & Creamy Sundaes charges $2.50 per sundae plus an additional $0.25 for each topping. Colder & Creamier Sundaes charges $1.50 per sundae plus an additional $0.50 for each topping. Determine the number of toppings for which both vendors charge the same amount. Explain which vendor is the better buy depending on the number of toppings Raja chooses. Let x represent the number of toppings on a sundae. Let y represent the cost (in dollars) for a sundae with x toppings. y 5 0.25x 1 2.50 Cold & Creamy Sundaes y 5 0.50x 1 1.50 Colder & Creamier Sundaes One possible solution path: Substitution Method: 0.25x 1 2.50 5 0.50x 1 1.50 y 5 0.25(4) 1 2.50 2.50 5 0.25x 1 1.50 y 5 1.00 1 2.50 1.00 5 0.25x y 5 3.50 45x © 2012 Carnegie Learning The solution is (4, 3.50). Both vendors charge $3.50 for a sundae with 4 toppings. If Raja wants fewer than 4 toppings, then Colder & Creamier Sundaes is the better buy. If Raja wants more than 4 toppings, Cold & Creamy Sundaes is the better buy. 6 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 439 439 4/23/12 12:05 PM Lesson 6.3 Skills Practice page 4 4. Marcus is selling t-shirts at the State Fair. He brings 200 shirts to sell. He has long-sleeve and short-sleeved T-shirts for sale. On the first day of the fair, he sells __ 1 of his long-sleeved T-shirts and __ 1 2 3 of his short-sleeved T-shirts for a total of 80 T-shirts sold. How many of each type of T-shirt did Marcus bring to the fair? Let x represent the number of long-sleeved T-shirts Marcus brings to the fair. Let y represent the number of short-sleeved T-shirts Marcus brings to the fair. __ x 1 y 5 200 1 1 x 1 y 5 80 3 2 __ One possible solution path: Linear Combinations Method: x 1 y 5 200 ( __ __ ) 6 1 x 1 1 y 5 80 2 3 x 1 y 5 200 3x 1 2y 5 480 22(x 1 y 5 200) 3x 1 2y 5 480 22x 2 2y 5 2400 ______________ 3x 1 2y 5 480 x 5 80 80 1 y 5 200 y 5 120 © 2012 Carnegie Learning The solution is (80, 120). Marcus brought 80 long-sleeved T-shirts and 120 short-sleeved T-shirts to the fair. 6 440 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 440 4/23/12 12:05 PM Lesson 6.3 Skills Practice Name page 5 Date 5. Alicia has a booth at the flea market where she sells purses and wallets. All of her purses are the same price and all of her wallets are the same price. The first hour of the day, she sells 10 purses and 6 wallets for a total of $193. The second hour, she sells 8 purses and 10 wallets for a total of $183. How much does Alicia charge for each purse and each wallet? Let x represent the charge for each purse. Let y represent the charge for each wallet. 10x 1 6y 5 193 8x 1 10y 5 183 One possible solution path: Linear Combinations Method: 5(10x 1 6y 5 193) 23(8x 1 10y 5 183) 50x 1 30y 5 965 224x 2 30y 5 2549 26x 5 416 x 5 16 10(16) 1 6y 5 193 160 1 6y 5 193 6y 5 33 y 5 5.5 © 2012 Carnegie Learning The solution is (16, 5.5). Alicia charges $16 for each purse and $5.50 for each wallet. 6 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 441 441 4/23/12 12:05 PM Lesson 6.3 Skills Practice page 6 6. Weston wants to buy a one-year membership to a golf course. Rolling Hills Golf Course charges a base fee of $200 and an additional $15 per round of golf. Majestic View Golf Course charges a base fee of $350 and an additional $10 per round of golf. Determine the number of rounds of golf for which both golf courses charge the same amount. Explain which golf course Weston should become a member at depending on the number of rounds he intends to play. Let x represent the number of rounds of golf played. Let y represent the cost of a one-year membership when x rounds of golf are played. y 5 15x 1 200 Rolling Hills Golf Course y 5 10x 1 350 Majestic View Golf Course One possible solution path: Substitution Method: 15x 1 200 5 10x 1 350 y 5 15(30) 1 200 15x 5 10x 1 150 5x 5 150 y 5 450 1 200 y 5 650 x 5 30 © 2012 Carnegie Learning The solution is (30, 650). Both golf courses charge $650 for a one-year membership when the member plays 30 rounds of golf. If Weston plans to play fewer than 30 rounds of golf, then he should become a member at Rolling Hills Golf Course. If Weston plans to play more than 30 rounds of golf, then he should become a member at Majestic View Golf Course. 6 442 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 442 4/23/12 12:05 PM Lesson 6.4 Skills Practice Name Date Which Is the Best Method? Using Graphing, Substitution, and Linear Combinations Problem Set Write a system of equations to represent each problem situation. Solve the system of equations using any method and answer any associated questions. 1. Jun received two different job offers to become a real estate sales agent. Dream Homes offered Jun a base salary of $20,000 per year plus a 2% commission on all real estate sold. Amazing Homes offered Jun a base salary of $25,000 per year plus a 1% commission on all real estate sold. Determine the amount of real estate sales in dollars for which both real estate companies will pay Jun the same amount. Explain which offer Jun should accept based on the amount of real estate sales he expects to have. Let x represent the amount of Jun’s real estate sales in dollars. Let y represent the yearly income when Jun has x dollars in real estate sales. y 5 0.02x 1 20,000 Dream Homes y 5 0.01x 1 25,000 Amazing Homes One possible solution path: Substitution Method: 0.02x 1 20,000 5 0.01x 1 25,000 y 5 0.02(500,000) 1 20,000 0.01x 1 20,000 5 25,000 y 5 10,000 1 20,000 y 5 30,000 © 2012 Carnegie Learning 0.01x 5 5000 x 5 500,000 The solution is (500,000, 30,000). Both real estate companies will pay Jun $30,000 per year for $500,000 in real estate sales. If Jun expects to sell less than $500,000 of real estate per year, then he should accept the offer from Amazing Homes. If Jun expects to sell more than $500,000 of real estate per year, then he should accept the offer from Dream Homes. Chapter 6 Skills Practice 8069_Skills_Ch06.indd 443 6 443 4/23/12 12:05 PM Lesson 6.4 Skills Practice page 2 2. Stella is trying to choose between two rental car companies. Speedy Trip Rental Cars charges a base fee of $24 plus an additional fee of $0.05 per mile. Wheels Deals Rental Cars charges a base fee of $30 plus an additional fee of $0.03 per mile. Determine the amount of miles driven for which both rental car companies charge the same amount. Explain which company Stella should use based on the number of miles she expects to drive. Let x represent the number of miles driven. Let y represent the total cost of a rental car when driven x miles. y 5 0.05x 1 24 y 5 0.03x 1 30 One possible solution path: Graphing Method: 50 45 Rental Car Cost (dollars) The solution is (300, 39). Both rental car companies charge $39 for a car driven 300 miles. If Stella expects to drive fewer than 300 miles, then she should use Speedy Trip Rental Cars. If Stella expects to drive more than 300 miles, then she should use Wheels Deals Rental Cars. Speedy Trip y 5 0.05x 1 24 y Wheels Deals y 5 0.03x 1 30 40 35 30 25 20 15 10 5 100 200 300 Miles Driven 400 x 500 © 2012 Carnegie Learning 0 6 444 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 444 4/23/12 12:05 PM Lesson 6.4 Skills Practice page 3 Name Date 3. Renee has two job offers to be a door-to-door food processor salesperson. Pro Process Processors offers her a base salary of $15,000 per year plus an additional $25 for each processor she sells. Puree Processors offers her a base salary of $18,000 per year plus an additional $21 for each processor she sells. Determine the number of food processors Renee would have to sell for both companies to pay her the same amount. Explain which job offer Renee should accept based on the number of food processors she expects to sell. Let x represent the number of food processors sold. Let y represent Renee’s yearly income when she sells x food processors. y 5 25x 1 15,000 Pro Process Processors y 5 21x 1 18,000 Puree Processors One possible solution path: Substitution Method: 25x 1 15,000 5 21x 1 18,000 y 5 25(750) 1 15,000 4x 1 15,000 5 18,000 y 5 18,750 1 15,000 4x 5 3000 y 5 33,750 x 5 750 © 2012 Carnegie Learning The solution is (750, 33,750). Both companies will pay Renee $33,750 for selling 750 food processors. If Renee expects to sell fewer than 750 food processors in one year, then she should accept the offer from Puree Processors. If Renee expects to sell more than 750 food processors in one year, then she should accept the offer from Pro Process Processors. 6 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 445 445 4/23/12 12:05 PM Lesson 6.4 Skills Practice page 4 4. Alex needs to rent a bulldozer. Smith’s Equipment Rentals rents bulldozers for a delivery fee of $600 plus an additional $37.50 per day. Robinson’s Equipment Rentals rents bulldozers for a delivery fee of $400 plus an additional $62.50 per day. Determine the number of rental days for which both rental companies charge the same amount. Explain which company Alex should choose based on the number of days he expects to rent a bulldozer. Let x represent the number of days Alex rents a bulldozer. Let y represent the cost to rent a bulldozer for x days. y 5 37.50x 1 600 Smith’s Equipment Rentals y 5 62.50x 1 400 Robinson’s Equipment Rentals One possible solution path: Graphing Method: y Robinson’s y 5 62.50x 1 400 1800 1600 1400 Smith’s y 5 37.50x 1 600 1200 1000 800 600 400 200 0 2 x 4 6 8 10 12 14 16 18 20 Number of Rental Days The solution is (8, 900). Both rental companies charge $900 to rent a bulldozer for 8 days. If Alex expects to use the bulldozer for fewer than 8 days, then he should choose Robinson’s Equipment Rental. If Alex expects to use the bulldozer for more than 8 days, then he should choose Smith’s Equipment Rental. 6 446 © 2012 Carnegie Learning Bulldozer Rental Cost (dollars) 2000 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 446 4/23/12 12:05 PM Lesson 6.4 Skills Practice page 5 Name Date 5. Serena has job offers from two car dealerships. Classic Cars offers her a base salary of $22,000 per year plus an additional 1% commission on all sales she makes. Sweet Rides offers her a base salary of $13,000 per year plus an additional 2.5% commission on all sales she makes. Determine the amount of car sales in dollars for which both dealerships will pay Serena the same amount. Explain which offer Serena should accept based on the amount of car sales she expects to have. Let x represent the amount (in dollars) of car sales. Let y represent the yearly income when Serena has x dollars in car sales. y 5 0.01x 1 22,000 Classic Cars y 5 0.025x 1 13,000 Sweet Rides One possible solution path: Substitution Method: 0.01x 1 22,000 5 0.025x 1 13,000 y 5 0.01(600,000) 1 22,000 y 5 6000 1 22,000 22,000 5 0.015x 1 13,000 9000 5 0.015x y 5 28,000 x 5 600,000 © 2012 Carnegie Learning The solution is (600,000, 28,000). Both dealerships will pay Serena $28,000 for $600,000 in car sales. If Serena expects to have fewer than $600,000 in car sales in one year, then she should accept the offer from Classic Cars. If Serena expects to have more than $600,000 in car sales in one year, then she should accept the offer from Sweet Rides. 6 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 447 447 4/23/12 12:05 PM Lesson 6.4 Skills Practice page 6 6. Dominique is trying to choose a satellite Internet service provider. Reliable Satellite charges customers a monthly fee of $26 plus an additional $0.30 per hour of online time. Super Satellite charges customers a monthly fee of $18 plus an additional $0.50 per hour of online time. Determine the number of hours of online time for which both providers charge the same amount. Explain which provider Dominique should choose based on the number of hours she expects to spend online each month. Let x represent the number of online hours. Let y represent the monthly service charge for x online hours. y 5 0.30x 1 26 Reliable Satellite y 5 0.50x 1 18 Super Satellite One possible solution path: Substitution Method: 0.30x 1 26 5 0.50x 1 18 y 5 0.30(40) 1 26 y 5 12 1 26 26 5 0.20x 1 18 8 5 0.20x x 5 40 y 5 38 © 2012 Carnegie Learning The solution is (40, 38). Each provider charges a total monthly fee of $38 for 40 hours of online time. If Dominique expects to spend fewer than 40 hours online each month, then she should choose Super Satellite. If Dominique expects to spend more than 40 hours online each month, then she should choose Reliable Satellite. 6 448 Chapter 6 Skills Practice 8069_Skills_Ch06.indd 448 4/23/12 12:05 PM
