Topics
11. Bivariate Data
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United States of America
Grade 8

11.04 Problem solving with linear models

Lesson
Practice

Introduction

We now know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, we can informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. We have also learned how to identify a line of best fit.

This time, we will use an equation of a linear model to solve problems in the context of the data and interpret the slope and intercept.

Problem solving with linear models

In the real world, it is not typical that two numerical variables are perfectly linear related. For data that are not exactly linear but that have a linear trend, we can draw a line through the data, determine the equation of the line of best fit and use this to make predictions and answer questions.

Examples

Example 1

Several cars underwent a brake test and their age, x (in years), was measured against their stopping distance, y (in meters). The scatter plot shows the results and a line of best fit:

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\text{Age}
10
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30
40
50
\text{Distance}
a

State the y-intercept based on the linear model.

Worked Solution
Create a strategy

On a graph, the y-intercept is the point where the line intersects the y-axis

Apply the idea

The line of best fit intersects the y-axis at 30.

The y-intercept is 30.

b

What does the y-intercept indicate in context?

Worked Solution
Create a strategy

The y-intercept is the value of y when the value of x is equal to 0.

Apply the idea

The stopping distance, y is 30 \text{ m} when the age, x is 0.

The stopping distance is 30 \text{ m} when the car is new.

c

Using two marked points on the line, find the slope of the line.

Worked Solution
Create a strategy

The slope of the line is the value of the ratio of change in y to change in x which is the same between any two points on a line.

Apply the idea

Identify two points on the line and use the formula:

m=\dfrac{y_2-y_1}{x_2-x_1}, where m is the slope of the line.

We can use the coordinates of the y-intercept, (0,30) and the point (10, 50) to find the slope of the line.

\displaystyle m\displaystyle =\displaystyle \dfrac{50-30}{10-0}Substitute the value of the coordinates of the points
\displaystyle =\displaystyle 2Evaluate

The slope, m=2.

d

Interpret the slope in the context.

Worked Solution
Create a strategy

The slope is the constant rate of change.

Apply the idea

Each increase of the age in years of the car is associated with an increase in the stopping distance by 2 meters.

e

State the equation of the line in the form y=mx+b.

Worked Solution
Create a strategy

The slope-intercept form of the equation of the line is y=mx +b, where m is the slope and b is the y-intercept.

Apply the idea

From the answers in (a) and (c), the slope is 2 and the y-intercept is 30.

The equation fo the line is:

y=2x+30

f

Estimate the stopping distance of a car that is 5 years-old.

Worked Solution
Create a strategy

The equation of the line of best fit can be used to make predictions and to answer questions.

Apply the idea

Using the equation of the line of best fit:

y=2x+30:

\displaystyle y\displaystyle =\displaystyle 2(5)+30Substitute the values needed.
\displaystyle y\displaystyle =\displaystyle 40Evaluate

The stopping distance of a 5 year-old car is 40 meters.

Idea summary

For data that is not exactly linear but has a linear trend, we can draw a line through the data, determine the equation of the line of best fit, and use this to make predictions and answer questions.

Outcomes

8.SP.A.3

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept

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