Beginning Algebra With Applications, Chapter 5, 5.4, Section 5.4, Problem 72

The table below shows the median weekly salary of a union member for various years.

$\begin{array}{|c|c|}
\hline\\
\text{Year} & \text{Weekly Salary} \\
& \text{(in dollars)} \\
\hline\\
2000 & 696 \\
\hline\\
2001 & 718 \\
\hline\\
2002 & 740 \\
\hline\\
2003 & 760 \\
\hline\\
2004 & 781 \\
\hline
\end{array} $

a. Determine the equation of the line between (2000, 696) and (2004, 781).

We let $(x_1, y_1) = (2000,696)$ and $(x_2, y_2) = (2004, 781)$

Using the slope of the line,


$
\begin{equation}
\begin{aligned}

m =& \frac{y_2 - y_1}{x_2 - x_1}
\\
\\
m =& \frac{781-696}{2004-2000}
\\
\\
m =& \frac{85}{4}

\end{aligned}
\end{equation}
$


Using the Point Slope Formula, where $\displaystyle m = \frac{85}{4}$ and $(x_1, y_1) = (2000,696)$


$
\begin{equation}
\begin{aligned}

y - y_1 =& m(x - x_1)
&&
\\
\\
y-696 =& \frac{85}{4} (x-2000)
&& \text{Substitute } m = \frac{85}{4} \text{ and } (x_1, y_1) = (2000,696)
\\
\\
y - 696 =& \frac{85}{4}x - 500
&& \text{Apply Distributive Property}
\\
\\
y =& \frac{85}{4}x - 500 + 696
&& \text{Simplify}
\\
\\
y =& \frac{85}{4}x + 196
&&

\end{aligned}
\end{equation}
$


b. What was the average annual rate of change in the median weekly salary for a union employee between 2000 and 2004?


$
\begin{equation}
\begin{aligned}

\text{average rate of change} =& \frac{\text{weekly salary from 2004 - weekly salary from 2000}}{2004-2000}
\\
\\
=& \frac{781-696}{2004-2000}
\\
\\
=& \frac{85}{4}
\\
\\
=& 21.25

\end{aligned}
\end{equation}
$


The average annual rate of change in the median weekly salary for a union employee between 2000 and 2004 is $\$ 21.25$.

c. Suppose the trend shown by the equation in part a were to continue, what would be the median weekly salary of a union worker in 2012?

Since the rate of change is linear, the slope is the same for any points. So we let $(x_1, y_1) = (2000, 696)$ and $(x_2, y_2) = (2012,n)$, $\displaystyle m = \frac{85}{4}$, then


$
\begin{equation}
\begin{aligned}

\frac{85}{4} =& \frac{n-696}{2012-2000}
&&
\\
\\
\frac{85}{4} =& \frac{n-696}{12}
&& \text{Multiply both sides by } 12
\\
\\
255 =& n-696
&& \text{Add } 696
\\
\\
n =& 951
&&

\end{aligned}
\end{equation}
$


The median weekly salary of a union worker in 2012 is $\$ 951$.