Chapter 8.6, Problem 23E

To determine

To find:

The equation of the line that is perpendicular to the given line and passes through $\left(0,1\right)$ .

Answer to Problem 23E

The equation of perpendicular lineto our given line is $y=4x+1$ .

Explanation of Solution

Given:

An equation: $y=-\frac{1}{4}x+3$ .

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 $-\frac{1}{4}$ .

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

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

So, the slope of perpendicular line to our given line is $4$ .

Now we will substitute the slope and y -intercept $\left(0,1\right)$ in slope-intercept form of equation as:

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

Therefore, the equation of perpendicular lineto our given line is $y=4x+1$ .