lesson 20a: solution sets to equations with two variables

Hart Interactive – Algebra 1 M1 Lesson 20A

ALGEBRA I

Lesson 20A: Solution Sets to Equations with Two Variables

Opening Exercise

1. The Math Club is having a Math Mystery Party. You are given the following clues about the location of the

party.

A. What are ALL the possible restaurants for the party based on the hints?

B. How did you decide whether or not an ordered pair was a solution to the equation?

New clues are on the way!

Alamo is at (3, 4). Foster’s Freeze is at (5, 10).

Baja Fresh is at (2, 3). Grand Panda is at (0, -10).

Chi-Chi’s Pizza is at (-1, -14). The Habit is at (3, 2).

Dippin Dots is at (0, 0). Islands is at (6, 0).

El Taco Llama is at (1, -6). Jazmine Thai Barbeque is at (4, -1).

The Math Mystery Party is at a restaurant on the line 4x – y = 10.


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ALGEBRA I

2. A. Discover as many additional solutions to the equation 4𝑥𝑥 − 𝑦𝑦 = 10 as possible. Consider the best way

to organize all the solutions you have found. Be prepared to share the strategies you used to find your

solutions.

B. Now, find five more solutions where one or more variables are negative numbers or non-integer

values. Be prepared to share the strategies you used to find your solutions.

C. How many ordered pairs (𝑥𝑥,𝑦𝑦) will be in the solution set of the equation 4𝑥𝑥 − 𝑦𝑦 = 10?

D. Create a visual representation of the solution set

by plotting each solution as a point (𝑥𝑥,𝑦𝑦) in the

coordinate plane.

E. Why does it make sense to represent the solution

to the equation 4𝑥𝑥 − 𝑦𝑦 = 10 as a line in the

coordinate plane?


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ALGEBRA I

3. Richard thinks of two numbers. The only clue he gives is

“The sum of two numbers is 10.” What are the numbers

Richard could be thinking of?

A. Create an equation using two variables to represent

this situation. Be sure to explain the meaning of each

variable.

B. List at least six solutions to the equation you created

in part (A).

C. Create a graph that represents the solution set to the

equation.

4. Gia had 10 songs in a playlist composed of songs from her two

favorite artists, Beyonce and Jennifer Lopez. How many songs did

she have by each one in the playlist?

A. Create an equation using two variables to represent this

situation. Be sure to explain the meaning of each

variable.

B. List at least three solutions to the equation you created in

part (a).

C. Create a graph that represents the solution set to the

equation.

5. Compare your solutions to Exercises 3 and 4. How are they alike? How are they different?


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ALGEBRA I

6. From Exercise 1 you should have a list of five possible restaurants for the Math Club Party. Here is a new

clue to help you find the correct location. Where is the party?

7. Graph each of the following equations.

A. y = -2 B. x = 3 C. y = 0

The point (3, -6) has a horizontal line going through it.

The equation of the horizontal line is y = -6.

What is the equation of a vertical line going through that point?

What other point of the five you found, does the vertical line go through?

That is the location of the Math Club Party!


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ALGEBRA I

Lesson Summary One way to graph a linear equation is to choose values for one variable and then determine the value for the other variable. Use number sense to help you choose appropriate values.

Graph 13 7

2y x+ = .

Vertical and Horizontal Lines

Graph Equation Characteristics

Vertical Lines

x = number

• No y-value

• Slope is undefined

Horizontal Lines

Y = number

• No x-value

• Slope is 0

Choose values for y

1

2ywhere will give an

integer value.

x y 2 2

3 -4


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ALGEBRA I

Homework Problem Set 1. Match each equation with its graph.

Explain your reasoning.

A. 𝑦𝑦 = 5𝑥𝑥 − 6

B. 𝑥𝑥 + 2𝑦𝑦 = −12

C. 2𝑥𝑥 + 𝑦𝑦 = 4

D. 𝑦𝑦 = 3𝑥𝑥 − 6

E. 𝑥𝑥 = −𝑦𝑦 − 4

2. Graph the solution set in the coordinate plane. Label at least two ordered pairs that are solutions on your

graph.

A. 10𝑥𝑥 + 6𝑦𝑦 = 100 B. 𝑦𝑦 = 9.5𝑥𝑥 + 20 C. 7𝑥𝑥 − 3𝑦𝑦 = 21 D. 𝑦𝑦 = 4(𝑥𝑥 + 10)

-10 -5 5 10

-10

-5

5

10

x

y

-10 -5 5 10

-10

-5

5

10

x

y 2 1

-10 -5 5 10

-10

-5

5

10

x

y 5

-10 -5 5 10

-10

-5

5

10

x

y

-10 -5 5 10

-10

-5

5

10

x

y 4 3


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Hart Interactive – Algebra M1 Lesson 20A ALGEBRA I

3. Mari and Lori are starting a business to make gourmet toffee. They gather

the following information from another business about prices for different

amounts of toffee. Which equation and which graph are most likely to

model the price, 𝑝𝑝, for 𝑥𝑥 pounds of toffee? Justify your reasoning.

Graph 1 Graph 2

Pounds

𝑥𝑥

Price, 𝑝𝑝,

for 𝑥𝑥

pounds

0.25 $3.60 0.81 $6.48

1 $7.20 1.44 $8.64

Equation A: 𝑝𝑝 = 5𝑥𝑥 + 2.2

Equation B: 𝑝𝑝 = 7.2√𝑥𝑥

4. Alexis realized that she could use linear equations to graph the first letter of her name. She came up with

the following three equations.

2 1

2 9

2

y xy xy

= += − +=

A. Graph her three equations and then highlight the “A”.

B. What should she add to her description so that only a part

of each line is drawn?

1 2 3 4

5

10

15

20

0 x

y

1 2 3 4

5

10

15

20

0 x

y


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Hart Interactive – Algebra M1 Lesson 20A ALGEBRA I

5. Treasure Hunt! Below is a map of Math City. Choose a location on the map and then write two or more

equations that would help someone determine your location.

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Math City

Legend

A = Algebra Airport

B = Binomial Bank

C = Coefficient Café

D = Delta the Dog Doctor

E = Ellipse Eye Exams

F = Flipping for Fish Store

G = GCF Groceries

A

B

C

D

E

F

G


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lesson 20a: solution sets to equations with two variables

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