Chapter 8.5, Problem 8E

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 8E

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

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

Explanation of Solution

Given:

An equation: $3x+2y=16$ .

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:

  $\begin{array}{l}3x+2y=16\left(\text{Given equation}\right)\\ 3x-3x+2y=16-3x\left(\text{Subtracting }3x\text{from both sides}\right)\\ 2y=-3x+16\\ \frac{2y}{2}=\frac{-3x}{2}+\frac{16}{2}\left(\text{Dividing both sides by }2\right)\\ y=-\frac{3}{2}x+8\end{array}$

We can see from our given equation that it is in slope-intercept form and its slope is $-\frac{3}{2}$ .

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

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

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

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