Tickets, Please Using Substitution to Solve a Linear System, Part 2

Transcription

1 Problem 1 Students write linear systems of equations representing several situations. Using substitution, they will solve each linear system and interpret the solution of the system in terms of each problem situation. In the last set of problems, students are given a linear system of equations and students solve each system. The first two systems and the last system have a unique solution, the third system has no solution, and the fourth system has an infinite number of solutions. Grouping Have students complete Question 1 with a partner. Then share the responses as a class. Learning Goals Tickets, Please Using Substitution to Solve a Linear System, Part 2 In this lesson, you will: Write a system of equations to represent a problem context. Solve a system of equations algebraically using substitution. Problem 1 Establishing Ticket Prices 1. The business manager for a band must make $236,000 from ticket sales to cover costs and make a reasonable profit. The auditorium where the band will play has 4000 seats, with 2800 seats on the main level and 1200 on the upper level. Attendees will pay $20 more for main-level seats. a. Write a system of equations with x representing the main-level seating and y representing the upper-level seating. 2800x y 5 236,000 x 5 y 1 20 Share Phase, Question 1 y 5 x 1 20 or x 5 y 1 20? Are all of the terms in one of the equations associated with money? Is 4000 seats used to write either equation? Is it easier to solve this equation for the value of x or the value of y? How is substitution used to solve this system of equations? b. Without solving the system of linear equations, interpret the solution. The solution will represent the cost, in dollars, of the main-level tickets and the upper-level tickets needed to make the targeted total sales. c. Solve the system of equations using the substitution method. 2800( y 1 20) y 5 236, y 1 56, y 5 236, y 1 56, , , , y 5 180,000 y 5 45 x x 5 65 The solution is (65, 45) Using Substitution to Solve a Linear System, Part Using Substitution to Solve a Linear System, Part 2 631

2 Grouping Have students complete Questions 2 through 4 with a partner. Then share the responses as a class. d. Interpret the solution of the system in terms of the problem situation. In order to make the targeted total sales, the cost of main-level seating will be $65 and the cost of upper-level seating will be $45. Share Phase, Question 2 x 1 y 5 20 or x 1 y 5 100? y 5 x 1 20 or x 5 y 1 20? Are all of the terms in one of the equations associated with the number of questions on the test? Are all of the terms in one of the equations associated with the number of points on the test? Is it easier to solve this equation for the value of x or the value of y? How is substitution used to solve this system of equations? 2. Ms. Ross told her class that tomorrow s math test will have 20 questions and be worth 100 points. The multiple-choice questions will be 3 points each and the open-ended response questions will be 8 points each. Determine how many multiple-choice and open-ended response questions will be on the test. a. Write a system of equations. Describe your variables. Let x represent the number of multiple-choice questions and y represent the number of open-ended response questions. x 1 y x 1 8y b. Without solving the system of linear equations, interpret the solution. The solution will represent the number of multiple-choice questions and the number of open-ended response questions on the 100-point test. c. Solve the system of equations using the substitution method. x y 3(20 2 y) 1 8y y 1 8y y 5 40 y 5 8 x x 5 12 (12, 8) d. Interpret the solution of the system in terms of the problem situation. There will be 12 multiple-choice questions and 8 open-ended response questions on the test. 632 Chapter 11 Systems of Equations 632 Chapter 11 Systems of Equations

3 Share Phase, Question 3 2x + 2y = 34 or 2x + 2y = 23? What term did you use to represent the cost of 2 drinks? 3 drinks? What term did you use to represent the cost of 1 pizza? 2 pizzas? Is it easier to solve one of the equations for the value of x or the value of y? How is substitution used to solve this system of equations? 3. Serena is ordering lunch from Tony s Pizza Parlor. John told her that when he ordered from Tony s last week, he paid $34 for two 16-inch pizzas and two drinks. Jodi told Serena that when she ordered one 16-inch pizza and three drinks, it cost $23. What is the cost of one 16-inch pizza and one drink? a. Write a system of equations. Describe your variables. Let x represent the cost of a 16-inch pizza and y represent the cost of a drink. 2x 1 2y 5 34 x 1 3y 5 23 b. Without solving the system of linear equations, interpret the solution. The solution will represent the cost of one 16-inch pizza and one drink. c. Solve the system of equations using the substitution method. x y 2(23 2 3y) 1 2y y y y 5 3 x 1 3(3) 5 23 x x 5 14 (14, 3) d. Interpret the solution of the system in terms of the problem situation. A 16-inch pizza costs $14 and a drink costs $ Using Substitution to Solve a Linear System, Part Using Substitution to Solve a Linear System, Part 2 633

4 Share Phase, Question 4 2x + 2y = 48 or 2x + 2y = 40? Is it easier to solve one of the equations for the value of x or the value of y? How is substitution used to solve this system of equations? 4. Ashley is working as ticket-taker at the arena. What should she tell the next person in line? Show your work and explain your reasoning. Student ticket 48 dollars, please. 40 dollars, please. Adult ticket??? 2x 1 2y 5 48 x 1 3y 5 40 x y x 1 3(8) (40 2 3y) 1 2y 5 48 x y 1 2y 5 48 x y y (16, 8) 3(16) 1 5(8) y Adult tickets cost $16 each and student tickets cost $8 each. The total cost of three adult and five student tickets will be $ Chapter 11 Systems of Equations 634 Chapter 11 Systems of Equations

5 Grouping Have students complete Question 5 with a partner. Then share the responses as a class. Share Phase, Question 5 Which was easiest? What do you suppose the graph of this system of linear equations looks like? Does this system of equations have a unique solution? If so, what is it? How do you know when a system of equations has a unique solution? How do you know when a system of equations has no solution? How do you know when a system of equations has an infinite number of solutions? 5. Solve each linear system of equations using the substitution method. Show your work. 2x 1 4y 5 4 a. x 2 2y 5 0 b. x 5 2y 2(2y) 1 4y x 1 4 ( 1 2) 5 4 2(1) 1 4 ( 1 2) 5 4 4y 1 4y 5 4 2x y 5 4 2x y x 5 1 The solution is ( 1, 1 x 5 2y 1 1 y x 1 1 2). x 5 2 ( 1 4 x 1 1 ) 1 1 y (6) 1 1 x x y x 5 3 y x 5 6 The solution is (6, 2.5). c. x 2 2y 5 4 x 2 2y 5 9 x 5 2y 1 4 2y y fi 9 There is no solution. It is important that you check your solution when you`re done Using Substitution to Solve a Linear System, Part Using Substitution to Solve a Linear System, Part 2 635

6 3x 1 2y 5 6 d. 1.5x 1 y 5 3 y x 1 3 3x 1 2(21.5x 1 3) 5 6 3x 2 3x There are an infinite number of solutions. e. 4x 1 3y x 5 2y ( 1 3 x ) 5 3(2y 1 1) 4(6y 1 3) 1 3y 5 27 x 5 6 ( 9) x 5 6y y y 5 27 x ). The solution is ( 19 3, 5 Check: 4 ( 19 3 ) 1 3 ( 5 9) Be prepared to share your solutions and methods. 27y 5 15 x y x x Chapter 11 Systems of Equations 636 Chapter 11 Systems of Equations