Chapter 8.5, Problem 29E

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 29E

The slope of perpendicular lineto our given line is $\frac{3}{2}$ .

The slope of parallel lineto our given line is $-\frac{2}{3}$ .

Explanation of Solution

Given:

A graph:

  

Concept used:

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

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:

First of all, we will find slope of our given line.

Let point $(0,3)=(x_1,y_1)\text{and (}3,1)=(x_2,y_2)$ . Upon substituting our given coordinates in slope formula, we will get:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{1-3}{3-0}\\ m=\frac{-2}{3}\\ m=-\frac{2}{3}\end{array}$

We can see that the slope of our given line is $-\frac{2}{3}$ . Since slope of two parallel lines is equal, therefore, the slope of parallel line to our given line would be $-\frac{2}{3}$ .

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

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

  $\begin{array}{l}-\frac{2}{3}\cdot m=-1\\ -\frac{2}{3}\cdot \left(-\frac{3}{2}\right)\cdot m=-1\cdot \left(-\frac{3}{2}\right)\left(\text{Multiplying both sides by }-\frac{3}{2}\right)\\ m=\frac{3}{2}\end{array}$

Therefore, the slope of perpendicular line to our given line is $\frac{3}{2}$ .