Intermediate Algebra: Connecting Concepts through Application

Published by Brooks Cole
ISBN 10: 0-53449-636-9
ISBN 13: 978-0-53449-636-4

Chapter 1 - Linear Functions - 1.5 Finding Equations of Lines - 1.5 Exercises - Page 75: 17

Answer

$y=8.5n+25$

Work Step by Step

Let $n$ be the number of shirts and $y$ be the corresponding amount of shirts. A linear equation that gives the cost, $y$, for $n$ shirts takes the form of $$ y=mn+b, $$ where $m$ is the slope and $b$ is the $y$-intercept. Since $10$ shirts cost \$$110$, this can be represented by the ordered pair $(n_1,y_1)=(10,110)$. Since $30$ shirts cost \$$280$, this can be represented by the ordered pair $(n_2,y_2)=(30,280)$. The formula for finding the slope, $m$, of the line passing through two points, $(n_1,y_1)$ and $(n_2,y_2)$ is given by $m=\frac{y_1-y_2}{n_1-n_2}$. That is, $$\begin{aligned} m&=\frac{y_1-y_2}{n_1-n_2} \\&= \frac{110-280}{10-30} \\&= \frac{-170}{-20} \\&= 8.5 .\end{aligned} $$ With $m=8.5$, the linear equation that gives the cost, $y$, for $n$ shirts takes the form of $$ y=8.5n+b .$$ Since the line passes through the point $(10,110)$, substitute $n=10$ and $y=110$ in the equation above to solve for $b$. That is, $$\begin{aligned} y&=8.5n+b \\ 110&=8.5(10)+b \\ 110&=85+b \\ 110-85&=b \\ b&=25 .\end{aligned} $$ With $b=25$, then the linear equation that gives the cost for $n$ shirts is $y=8.5n+25$.
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