# Solution Sets to Equations with Two Variables

Examples, videos, and solutions to help Algebra I students learn how to solve equations with two variables.

### New York State Common Core Math Algebra I, Module 1, Lesson 20

Lesson 20 Student Outcomes

Students recognize and identify solutions to two-variable equations. They represent the solution set graphically. They create two variable equations to represent a situation. They understand that the graph of the line ax + by = c is a visual representation of the solution set to the equation ax + by = c.

Lesson 20 Summary

An ordered pair is a solution to a two variable equation when each number substituted into its corresponding variable makes the equation a true number sentence. All of the solutions to a two variable equation are called the solution set.

Each ordered pair of numbers in the solution set of the equation corresponds to a point on the coordinate plane. The set of all such points in the coordinate plane is called the graph of the equation.

Exercises

1. a. Circle all the ordered pairs (𝑥, 𝑦) that are solutions to the equation 4𝑥 − 𝑦 = 10.
(3,2) (2, 3) (−1,−14) (0,0) (1,−6)
(5,10) (0,−10) (3,4) (6,0) (4,−1)
b. How did you decide whether or not an ordered pair was a solution to the equation?
2. a. Discover as many additional solutions to the equation 4𝑥 − 𝑦 = 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 (𝑥, 𝑦) will be in the solution set of the equation 4𝑥 − 𝑦 = 10?
d. Create a visual representation of the solution set by plotting each solution as a point (𝑥, 𝑦) in the coordinate plane.
e. Why does it make sense to represent the solution to the equation 4𝑥 − 𝑦 = 10 as a line in the coordinate plane?
3. The sum of two numbers is 25. What are the numbers?
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 25 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?

Exit Ticket

1. The Math Club sells hot dogs at a school fundraiser. The club earns \$108 and has a combination of five-dollar and one-dollar bills in its cash box. Possible combinations of bills are listed in the table below.
a. Find one more combination of ones and fives that totals \$108.
b. The equation 5x + 1y = 108 represents this situation. A graph of the line y = -x + 108 is shown. Verify that each ordered pair in the table lies on the line.
c. What is the meaning of the variables (x and y) and the numbers (1, 5, and 108) in the equation ?
d. Does the graph make sense in this context?

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