Graphs of Linear Equations in Two Variables

Examples, solutions, worksheets, and videos to help Grade 8 students learn that any constant rate problem can be described by a linear equation in two variables where the slope of the graph is the constant rate.

New York State Common Core Math Grade 8, Module 4, Lesson 22

Worksheets for Grade 8

Lesson 22 Student Outcomes

• Students know that any constant rate problem can be described by a linear equation in two variables where the slope of the graph is the constant rate.
• Students compare two different proportional relationships represented by graphs, equations, and tables to determine which has a greater rate of change.

Lesson 22 Student Summary

Problems involving constant rate can be expressed as linear equations in two variables.
When given information about two proportional relationships, determine which is less or greater by comparing their slopes (rates of change).

Lesson 22 Opening Exercise

Example 1:
Erika set her stopwatch to zero and switched it on at the beginning of her walk. She walks at a constant speed of miles per hour. Present this situation as an equation, table of values, and a graph.
What is the definition of constant rate?
What will the two variables represent?

Example 2:
A faucet leaks at a constant rate of 7 gallons per hour. Suppose y gallons leak in x hours from 6:00 a.m. Express the situation as a linear equation in two variables.
Another faucet leaks at a constant rate and the table below shows the number of gallons y, that leak in x hours. Determine the rate at which this faucet leaks? Which faucet has the worse leak?

Example 3:
The graph below represents the constant rate at which Train A travels. What is the constant rate of travel for Train A?
Train B travels at a constant rate. The train travels at an average rate of miles every one and a half hours. Which train is traveling at a greater speed? Explain.
Why do you think the strategy of comparing each rate of change allows us to determine which train is travelling at a greater speed?

Example 4:
The graph below represents the constant rate at which Kristina can paint.
Her sister Tracee paints at an average rate of square feet in minutes. Assuming Tracee paints at a constant rate, determine which sister paints faster.
How does the slope provide the information we need to answer the question about which sister paints faster?

Example 5:
The graph below represents the constant rate of watts of energy produced from a single solar panel produced by Company A.
Company B offers a solar panel that produces energy at an average rate of watts in hours. Assuming solar panels produce energy at a constant rate, determine which company produces more efficient solar panels (solar panels that produce more energy per hour).

Exercises

1. Peter paints a wall at a constant rate of 2 square-feet per minute. Assume he paints an area y, in square feet after x minutes.
   a. Express this situation as a linear equation in two variables.
   b. Graph the linear equation.
   c. Using the graph or the equation, determine the total area he paints after 8 minutes, 1 1/2 hours, and 2 hours.
   Note that the units are in minutes and hours.
2. The graph below represents Nathan’s constant rate of walking.
   a. Nicole just finished a 5 mile walkathon. It took her 1.4 hours. Assume she walks at a constant rate. Let y represent the distance Nicole walks in x hours. Describe Nicole’s walking at a constant rate as a linear equation in two variables.
   b. Who walks at a greater speed? Explain.
3. a. Susan can type 4 pages of text in 10 minutes. Assuming she types at a constant rate, write the linear equation that represents the situation.
   b. The table of values below represents the number of pages that Anne can type, y, in x minutes. Assume she types at a constant rate.
   Who types faster? Explain.
4. a. Phil can build 3 birdhouses in 3 days. Assuming he builds birdhouses at a constant rate, write the linear equation that represents the situation.
   b. The graph represents Karl’s constant rate of building the same kind of birdhouses. Who builds birdhouses faster? Explain.
5. Explain your general strategy for comparing proportional relationships.

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