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

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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.

Share Phase,

Question 1

•

Which equation makes more sense in this situation,

y5x1 20 or x5y1 20? Explain.

•

Are all of the terms in one of the equations associated with money?

•

Is 4000 seats used to write

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Learning Goals

In this lesson, you will:

 Write a system of equations to represent a problem context.

 Solve a system of equations algebraically using substitution.

Tickets, Please

Using Substitution to Solve

a Linear System, Part 2

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.



2800 _x₅_yx₁1 1200₂₀ y5 236,000

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( y1 20) 1 1200y5 236,000 2800y1 56,000 1 1200y5 236,000 4000y1 56,000 2 56,000 5 236,000 2 56,000 4000y5 180,000 y5 45 x5 45 1 20

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632

•

Chapter 11 Systems of Equations

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Grouping

Have students complete Questions 2 through 4 with a partner. Then share the responses as a class.

Share Phase,

Question 2

•

x1y5 20 or x1y5 100? Explain.

•

y5x1 20 or x5y1 20? Explain.

•

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?

•

Did you solve this system of linear equations for the value of x or the value of y?

•

Is it easier to solve this equation for the value of x or the value of y? Explain.

•

How is substitution used to solve this system of equations?

632 • Chapter 11 Systems of Equations

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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.

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 ₃_x1₁y₈5_y 20 ₅₁₀₀

The solution will represent the number of multiple-choice questions and the number of open-ended response questions on the 100-point test.

x5 20 2y 3(20 2y) 1 8y5 100 60 2 3y1 8y5 100 5y5 40 y5 8 x1 8 5 20 x5 12 (12, 8)

There will be 12 multiple-choice questions and 8 open-ended response questions on the test.

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Share Phase,

Question 3

•

Which equation makes more sense in this situation, 2x + 2y = 34 or 2x + 2y = 23? Explain.

•

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? Explain.

•

How is substitution used to solve this system

of equations?

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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.



2 _xx1₁ 2₃y_y5₅ 34 ₂₃

The solution will represent the cost of one 16-inch pizza and one drink.

x5 23 2 3y 2(23 2 3y) 1 2y5 34 46 2 4y5 34 24y5212 y5 3 x1 3(3) 5 23 x1 9 5 23 x5 14 (14, 3)

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634

• Share Phase,

Question 4

•

Which equation makes more sense in this situation, 2x + 2y = 48 or 2x + 2y = 40? Explain.

•

Is it easier to solve one of the equations for the value of x

or the value of y? Explain.

•

How is substitution used to solve this system of equations?

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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.

48 dollars, please. 40 dollars, please. ? ? ? ADMIT ONE ADMIT ONE ADMIT ONE ADMIT

ONE ADMITONE ADMIT

ONE

ADMIT

ONE ADMIT_ONE ADMIT_ONE ADMIT_ONE ADMIT_ONE

ADMIT ONE ADMIT ONE ADMIT ONE ADMIT ONE ADMIT

ONE ADMITONE ADMITONE

Student ticket _{Adult ticket}



2 _xx₁1₃ 2_yy₅5₄₀ 48 x5 40 2 3y x1 3(8) 5 40 2(40 2 3y) 1 2y5 48 x1 24 5 40 80 2 6y1 2y5 48 x5 16 80 2 4y5 48 24y5232 (16, 8) 3(16) 1 5(8) 5 48 1 40 y5 8 5 88

Adult tickets cost $16 each and student tickets cost $8 each. The total cost of three adult and five student tickets will be $88.

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Grouping

Have students complete Question 5 with a partner. Then share the responses as a class.

Share Phase,

Question 5

•

Did you solve this system of linear equations for the value of x or the value of y? Which was easiest? Explain.

•

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?

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5. Solve each linear system of equations using the substitution method. Show your work. a.



2x 1 4y 5 4 _{x 2 2y 5 0} x5 2y 2(2y) 1 4y5 4 2 x1 4

(

1 __ ₂

)

5 4 2(1) 1 4

(

1 __ ₂

)

5 4 4y1 4y5 4 2x1 2 5 4 2 1 2 5 4 8y5 4 2x5 2 4 5 4 ✔ y5 1 __ ₂x5 1 The solution is

(

1, 1 __ ₂

)

. b.



x 5 2y 1 1 y 5 1 __ 4 x 1 1 x5 2

(

1 __ ₄ x1 1

)

1 1 y5 1 __ ₄ (6) 1 1 x5 1 __ ₂ x1 2 1 1 y5 1.5 1 1 1 __ ₂ x5 3 y5 2.5 x5 6 The solution is (6, 2.5). c.



x 2 2y 5 4 x 2 2y 5 9 x5 2y1 4 2y1 4 2 2y5 9 4 ﬁ 9 There is no solution.

It is

important that

you check your

solution when

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636

•

636 • Chapter 11 Systems of Equations d.



3x 1 2y 5 6 1.5x 1 y 5 3 y521.5x1 3 3x1 2(21.5x1 3) 5 6 3x2 3x1 6 5 6 6 5 6

There are an infinite number of solutions.

e.



4x 1 3y 5 27 1 __ 3 x 5 2y 1 1 3

(

1 __ ₃ x

)

5 3(2y1 1) 4(6y1 3) 1 3y5 27 x5 6

(

5 __ ₉

)

1 3 x5 6y1 3 24y1 12 1 3y5 27 x5 10 ___ ₃ 1 3 27y5 15 x5 3 1 10 ___ ₃ y5 15 ___ ₂₇ 5 5 __ ₉ x5 9 __ ₃ 1 10 ___ ₃ x5 19 ___ ₃ The solution is

(

19 ___ ₃ , 5 __ ₉

)

. Check: 4

(

19 ___ ₃

)

1 3

(

5 __ ₉

)

5 27 76 ___ 3 1 5 __ 3 5 27 81 ___ ₃ 5 27 27 5 27