Chapter 8.5, Problem 25E

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 25E

The slope of perpendicular lineto our given line is $-\frac{5}{4}$ .

The slope of parallel lineto our given line is $\frac{4}{5}$ .

Explanation of Solution

Given:

An equation: $4x-5y=30$ .

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:

  $\begin{array}{l}4x-5y=30\left(\text{Given equation}\right)\\ 4x-4x-5y=30-4x\left(\text{Subtracting }4x\text{from both sides}\right)\\ -5y=-4x+30\\ \frac{-5y}{-5}=\frac{-4x}{-5}+\frac{30}{-5}\left(\text{Dividing both sides by }-5\right)\\ y=\frac{4}{5}x-6\end{array}$

We can see from our given equation that it is in slope-intercept form and its slope is $\frac{4}{5}$ .

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

  $\begin{array}{l}\frac{4}{5}\cdot m=-1\\ \left(\frac{5}{4}\right)\cdot \frac{4}{5}\cdot m=-1\cdot \left(\frac{5}{4}\right)\left(\text{Multiplying both sides by }\frac{5}{4}\right)\\ m=-\frac{5}{4}\end{array}$

So, the slope of perpendicular line to our given line is $-\frac{5}{4}$ .

We can see that the slope of our given line is $\frac{4}{5}$ . Since slope of two parallel lines is equal, therefore, the slope of parallel line to our given line would be $\frac{4}{5}$ .