Chapter 8.7, Problem 27E

To determine

To write:

A linear function that satisfies the given conditions.

Answer to Problem 27E

The function $r(x)=\frac{2}{3}x-1$ satisfies the given conditions.

Explanation of Solution

Given:

  $r(-9)=-7,r(0)=-1$

Concepts used:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1},\text{where,}\\ m=\text{Slope of line,}\\ y_2-y_1=\text{Difference between two }y\text{-coordinates}\\ x_2-x_1=\text{Difference between two }x\text{-coordinates of same }y-\text{coordinates}\end{array}$

The equation of a line in slope-intercept form is $y=mx+b$ , where m represents slope of line and b represents the y -intercept.

Calculation:

We have been given two points on a line that are $(-9,-7)\text{and }(0,-1)$ .

Now we will find the slope of line passing through the given points using slope formula.

Let point $(-9,-7)=(x_1,y_1)\text{and }(0,-1)=(x_2,y_2)$ . Upon substituting our given coordinates in slope formula, we will get:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-1-(-7)}{0-(-9)}\\ m=\frac{-1+7}{0+9}\\ m=\frac{6}{9}\\ m=\frac{2}{3}\end{array}$

To write our function, we will substitute the value of slope and the y -intercept $(0,-1)$ in slope-intercept form of equation as shown below:

  $\begin{array}{l}y=mx+b\left(\text{Slope-intercept form of equation}\right)\\ y=\frac{2}{3}\cdot x+(-1)\left(\text{Substituting }m=\frac{2}{3}\text{ and }b=-1\right)\\ y=\frac{2}{3}x-1\\ r(x)=\frac{2}{3}x-1\left(\text{Writing in function form}\right)\end{array}$

Therefore, the function $r(x)=\frac{2}{3}x-1$ satisfies the given conditions.