# Illustrative Mathematics Grade 8, Unit 3, Lesson 12: Solutions to Linear Equations

Learning Targets:

• I know that the graph of an equation is a visual representation of all the solutions to the equation.
• I understand what the solution to an equation in two variables is.

Related Pages
Illustrative Math

#### Lesson 12: Solutions to Linear Equations

Let’s think about what it means to be a solution to a linear equation with two variables in it.

Illustrative Math Unit 8.3, Lesson 12 (printable worksheets)

#### Lesson 12 Summary

The following diagram shows how the graph of an equation is a visual representation of all the solutions to the equation.

#### Lesson 12.1 Estimate Area

Which one doesn’t belong?

#### Lesson 12.2 Apples and Oranges

At the corner produce market, apples cost \$1 each and oranges cost \$2 each.

1. Find the cost of:
​a. 6 apples and 3 oranges
b. 4 apples and 4 oranges
c. 5 apples and 4 oranges
d. 8 apples and 2 oranges
2. Noah has \$10 to spend at the produce market. Can he buy 7 apples and 2 oranges? Explain or show your reasoning.
3. What combinations of apples and oranges can Noah buy if he spends all of his \$10?
4. Use two variables to write an equation that represents \$10-combinations of apples and oranges. Be sure to say what each variable means.
5. What are 3 combinations of apples and oranges that make your equation true? What are three combinations of apples and oranges that make it false?

#### Are you ready for more?

1. Graph the equation you wrote relating the number of apples and the number of oranges.
2. What is the slope of the graph? What is the meaning of the slope in terms of the context?

The slope is -1/2.
The slope means that one orange is worth 2 apples.

3. Suppose Noah has \$20 to spend. Graph the equation describing this situation. What do you notice about the relationship between this graph and the earlier one?

#### Lesson 12.3 Solutions and Everything Else

You have two numbers. If you double the first number and add it to the second number, the sum is 10.

1. Let x represent the first number and let represent the second number. Write an equation showing the relationship between x, y, and 10.
2. Draw and label a set of x- and y-axes. Plot at least five points on this coordinate plane that make the statement and your equation true. What do you notice about the points you have plotted?
3. List ten points that do not make the statement true. Using a different color, plot each point in the same coordinate plane. What do you notice about these points compared to your first set of points?

#### Lesson 12 Practice Problems

1. Select all of the ordered pairs (x,y) that are solutions to the linear equation 2x + 3y = 6.
2. The graph shows a linear relationship between x and y. x represents the number of comic books Priya buys at the store, all at the same price, and y represents the amount of money (in dollars) Priya has after buying the comic books.
a. Find and interpret the x- and y-intercepts of this line.
b. Find and interpret the slope of this line.
c. Find an equation for this line.
d. If Priya buys 3 comics, how much money will she have remaining?
3. Match each equation with its three solutions.
4. A container of fuel dispenses fuel at the rate of 5 gallons per second. If y represents the amount of fuel remaining in the container, and x represents the number of seconds that have passed since the fuel started dispensing, then x and y satisfy a linear relationship.
In the coordinate plane, will the slope of the line representing that relationship have a positive, negative, or zero slope? Explain how you know.
5. A sandwich store charges a delivery fee to bring lunch to an office building. One office pays \$33 for 4 turkey sandwiches. Another office pays \$61 for 8 turkey sandwiches. How much does each turkey sandwich add to the cost of the delivery? Explain how you know.

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.