Chapter 8.7, Problem 40E

To determine

To find:

The equation of the line that is perpendicular to the given line and passes through $\left(0,-4\right)$ .

Answer to Problem 40E

The equation of perpendicular lineto our given line is $y=-\frac{1}{6}x-4$ .

Explanation of Solution

Given:

An equation: $y=6x+10$ .

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.

The equation of a line in slope-intercept form is $y=mx+b$ , where m represents slope of line and b represents the y -intercept.

Calculation:

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

We can see from our given equation that it is in slope-intercept form and its slope is $6$ .

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

  $\begin{array}{l}6\cdot m=-1\\ \frac{6\cdot m}{6}=\frac{-1}{6}\text{ (Dividing both sides by }6\text{)}\\ m=-\frac{1}{6}\end{array}$

So, the slope of perpendicular line to our given line is $-\frac{1}{6}$ .

Now we will substitute the slope and y -intercept $\left(0,-4\right)$ in slope-intercept form of equation as:

  $\begin{array}{l}y=mx+b\left(\text{Slope-intercept form of equation}\right)\\ y=-\frac{1}{6}\cdot x+(-4)\left(\text{Substituting }m=-\frac{1}{6}\text{ and }b=-4\right)\\ y=-\frac{1}{6}x-4\end{array}$

Therefore, the equation of perpendicular lineto our given line is $y=-\frac{1}{6}x-4$ .