Chapter 8.6, Problem 6E

To determine

To find:

The equation of the line that is perpendicular to the given line and passes through $\left(0,-7\right)$ .

Answer to Problem 6E

The equation of perpendicular lineto our given line is $y=-\frac{1}{2}x-7$ .

Explanation of Solution

Given:

An equation: $y=2x-11$ .

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.

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 $2$ .

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

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

So, the slope of perpendicular line to our given line is $-\frac{1}{2}$ .

Now we will substitute the slope and coordinates of point $\left(0,-7\right)$ in point-slope form of equation as:

  $\begin{array}{l}y-y_1=m\left(x-x_1\right)\left(\text{Point-slope form of equation}\right)\\ y-(-7)=-\frac{1}{2}\left(x-0\right)\left(\text{Substituting given values}\right)\\ y+7=-\frac{1}{2}x\\ y+7-7=-\frac{1}{2}x-7\\ y=-\frac{1}{2}x-7\end{array}$

Therefore, the equation of perpendicular lineto our given line is $y=-\frac{1}{2}x-7$ .