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2.4: Find y given x and the Equation of a Line

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Learning Outcomes

1. Find the value of y given x and the equation of a line.
2. Use a line to make predictions.

A line can be thought of as a function, which means that if a value of $$x$$ is given, the equation of the line produces exactly one value of $$y$$; This is particularly useful in regression analysis where the line is used to make a prediction of one variable given the value of the other variable.

Example $$\PageIndex{1}$$

Consider the line with equation:

$$y=3x-4$$

Find the value of $$y$$ when $$x$$ is 5.

Solution

Just replace the variable $$x$$ with the number 5 in the equation and perform the arithmetic:

$$y\:=\:3\left(5\right)-4=15-4\:=11$$

Example $$\PageIndex{2}$$

A survey was done to look at the relationship between a woman's height, $$x$$ and the woman's weight, $$y$$. The equation of the regression line was found to be:

$$y=-220+5.5x$$

Use this equation to estimate the weight in pounds of a woman who is 5' 2" (62 inches) tall.

Solution

Just replace the variable $$x$$ with the number 62 in the equation and perform the arithmetic:

$$y\:=\:-220+5.5\left(62\right)$$

We can put this into a calculator or computer to get:

$$y\:=\:121$$

Therefore, our best prediction for the weight of a woman who is 5' 2'' tall is that she is 121 lbs.

Exercise

A biologist has collected data on the girth (how far around) of pine trees and the pine tree's height. She found the equation of the regression line to be:

$$y=1.3+2.7x$$

Where the girth, $$x$$, is measured in inches and the height, $$y$$, is measured in feet. Use the regression line to predict the height of a tree with girth 28 inches.

https://youtu.be/cS95PlUKZ6I