Chapter 8.5, Problem 28E

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 28E

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

The slope of parallel lineto our given line is $3$ .

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 $(1,1)=(x_1,y_1)\text{and (}2,4)=(x_2,y_2)$ . Upon substituting our given coordinates in slope formula, we will get:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{4-1}{2-1}\\ m=\frac{3}{1}\\ m=3\end{array}$

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

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:

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

So, the slope of perpendicular line to our given line is $-\frac{1}{3}$ .