Chapter 8.5, Problem 27E

To determine

To find:

The slope of a parallel line and the slope of a perpendicular line for the line with the given equation.

Answer to Problem 27E

The slope of perpendicular lineto our given line is $-3$ .

The slope of parallel lineto our given line is $\frac{1}{3}$ .

Explanation of Solution

Given:

An equation: $x=3y-7$ .

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.

We also know that the slopes of two parallel lines are equal.

Calculation:

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

Let us convert our given equation in slope-intercept form as shown below:

  $\begin{array}{l}x=3y-7\left(\text{Given equation}\right)\\ 3y-7=x\left(\text{Switching sides}\right)\\ 3y-7+7=x+7\left(\text{Adding }7\text{ on both sides}\right)\\ 3y=x+7\\ \frac{3y}{3}=\frac{x}{3}+\frac{7}{3}\left(\text{Dividing both sides by }3\right)\\ y=\frac{1}{3}x+\frac{7}{3}\end{array}$

We can see from our given equation that it is in slope-intercept form and its slope is $\frac{1}{3}$ .

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

  $\begin{array}{l}\frac{1}{3}\cdot m=-1\\ 3\cdot \frac{1}{3}\cdot m=-1\cdot 3\left(\text{Multiplying both sides by }3\right)\\ m=-3\end{array}$

So, the slope of perpendicular line to our given line is $-3$ .

We can see that the slope of our given line is $\frac{1}{3}$ . Since slope of two parallel lines is equal, therefore, the slope of parallel line to our given line would be $\frac{1}{3}$ .