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

11.6.3: 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 onumber ]

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:=:3left(5 ight)-4=15-4:=11 onumber ]

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 onumber ]

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+5.5left(62 ight) onumber ]

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

[y:=:121 onumber ]

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 onumber ]

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.