Find y given x and the Equation of a Line
- Page ID
- 4721
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\ket}[1]{\left| #1 \right>}\)
\(\newcommand{\bra}[1]{\left< #1 \right|}\)
\(\newcommand{\braket}[2]{\left< #1 \vphantom{#2} \right| \left. #2 \vphantom{#1} \right>}\)
\(\newcommand{\braopket}[3]{\left< #1 \vphantom{#2}\vphantom{#3} \right| #2 \vphantom{#1}\vphantom{#3} \left| #3 \vphantom{#1}\vphantom{#2} \right>}\)
\(\newcommand{\qmvec}[1]{\mathbf{\vec{#1}}}\)
\(\newcommand{\op}[1]{\hat{\mathbf{#1}}}\)
\(\newcommand{\expect}[1]{\langle #1 \rangle}\)
\(\newcommand{\dfn}[1]{\emph{\textbf{#1}}}\)
Learning Outcomes
- Find the value of y given x and the equation of a line.
- 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:
\[y=3x-4\nonumber \]
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:
\[y\:=\:3\left(5\right)-4=15-4\:=11\nonumber \]
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:
\[y=-220+5.5x\nonumber \]
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:
\[y\:=\:-220+5.5\left(62\right)\nonumber \]
We can put this into a calculator or computer to get:
\[y\:=\:121\nonumber \]
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:
\[y=1.3+2.7x\nonumber \]
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.