Chapter 4, Problem 48SGR

To determine

To Find:

Write an equation in slope-intercept form for the line of fit for the given situation.

Answer to Problem 48SGR

The required equation of line is :

  $y=210x+230$

Explanation of Solution

Given Information:

A scatter plot of data compares the number of years since a business has opened and its annual number of sales.It contains the ordered pairs (2,650) and (5,1280).

Formula Used:

The slope-intercept form of a line that passes through points $(x_1,y_1)\&(x_2,y_2)$

• is given by:

  $\begin{array}{l}slope-\mathrm{int}ercept\_form:\\ y=mx+b\\ m=\frac{y_2-y_1}{x_2-x_1}\end{array}$

• Point-slope form for a line that passes through point $(x_1,y_1)$ and has slope m :

  $\begin{array}{l}y-y_1=m(x-x_1)\\ where\\ (x_1,y_1)=po\mathrm{int}\_through\_which\_line\_passes\\ m=slope\end{array}$

Calculation:

The required line passes through points (2,650) and (5,1280).

1. Finding slope of line :

  $\begin{array}{l}(x_1,y_1)=(2,650)\\ (x_2,y_2)=(5,1280)\\ m=\frac{y_2-y_1}{x_2-x_1}=\frac{1280-650}{5-2}=\frac{630}{3}=210\end{array}$

2. Use the slope and any one of the two points to fine the y -intercept : $\begin{array}{l}m=210\\ (x,y)=(2,650)\\ slope-intercept\_form:\\ y=mx+b\\ y=210x+b\mathrm{..........................}[m=210]\\ 650=210(2)+b\mathrm{...................}[(x,y)=(2,650)]\\ 650=420+b\mathrm{.........................}[simplify]\\ 650-420=420+b-\mathrm{420......}[subtract"420"each\_side]\\ 230=b\end{array}$
3. Write the equation in slope-intercept form :

$\begin{array}{l}y=mx+b\\ m=210;b=230\\ y=210x+230\end{array}$

The required equation of line is :

  $y=210x+230$