Lesson 4

Introduction to Linear Relationships

Let’s explore some relationships between two variables.

Problem 1

A restaurant offers delivery for their pizzas. The total cost is a delivery fee added to the price of the pizzas. One customer pays $25 to have 2 pizzas delivered. Another customer pays $58 for 5 pizzas. How many pizzas are delivered to a customer who pays $80?

Problem 2

To paint a house, a painting company charges a flat rate of $500 for supplies, plus $50 for each hour of labor.

  1. How much would the painting company charge to paint a house that needs 20 hours of labor? A house that needs 50 hours?
  2. Draw a line representing the relationship between \(x\), the number of hours it takes the painting company to finish the house, and \(y\), the total cost of painting the house. Label the two points from the earlier question on your graph.

    quadrant 1 grid. horizontal axis, hours, scale 0 to 60, by 10's. vertical axis, dollar, 0 to 4,000, by 1,000. 
  3. Find the slope of the line. What is the meaning of the slope in this context?

Problem 3

Tyler and Elena are on the cross country team.

Tyler's distances and times for a training run are shown on the graph.

graph, horizontal axis, distance in miles, scale 0 to 4, by 1's. vertical axis, time in minutes, scale 0 to 190, by 2's.

Elena’s distances and times for a training run are given by the equation \(y=8.5x\), where \(x\) represents distance in miles and \(y\) represents time in minutes.

  1. Who ran farther in 10 minutes? How much farther? Explain how you know.
  2. Calculate each runner's pace in minutes per mile.
  3. Who ran faster during the training run? Explain or show your reasoning.
(From Unit 5, Lesson 3.)

Problem 4

Write an equation for the line that passes through \((2,5)\) and \((6,7)\).

(From Unit 2, Lesson 17.)