Chapter 8.6, Problem 22E

To determine

To find:

The equation of the line that is perpendicular to the given line and passes through $\left(0,-5\right)$ .

Answer to Problem 22E

The equation of perpendicular lineto our given line is $y=-x-5$ .

Explanation of Solution

Given:

An equation: $y=x-7$ .

Concept used:

We know that the slope of a perpendicular line to a given line is negative reciprocal of the slope of the given line. The product of slopes of two perpendicular lines is -1.

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:

Let mbe the slope of perpendicular line to our given line.

We can see from our given equation that it is in slope-intercept form and its slope is $1$ .

We will find the slope of perpendicular line to our given line as shown below:

  $\begin{array}{l}1\cdot m=-1\\ \frac{1\cdot m}{1}=\frac{-1}{1}\text{ (Dividing both sides by }1\text{)}\\ m=-1\end{array}$

So, the slope of perpendicular line to our given line is $-1$ .

Now we will substitute the slope and y -intercept $\left(0,-5\right)$ in slope-intercept form of equation as:

  $\begin{array}{l}y=mx+b\left(\text{Slope-intercept form of equation}\right)\\ y=-1\cdot x+(-5)\left(\text{Substituting }m=-1\text{ and }b=-5\right)\\ y=-x-5\end{array}$

Therefore, the equation of perpendicular lineto our given line is $y=-x-5$ .