Chapter 8.7, Problem 41E

To determine

To find:

The equation of the line that is perpendicular to the given line and passes through $\left(0,3\right)$ .

Answer to Problem 41E

The equation of perpendicular lineto our given line is $y=\frac{9}{5}x+3$ .

Explanation of Solution

Given:

An equation: $y=-\frac{5}{9}x-1$ .

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 $-\frac{5}{9}$ .

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

  $\begin{array}{l}-\frac{5}{9}\cdot m=-1\\ -\frac{5}{9}\cdot \left(-\frac{9}{5}\right)\cdot m=-1\cdot \left(-\frac{9}{5}\right)\left(\text{Multiplying both sides by }\cdot \left(-\frac{9}{5}\right)\right)\\ m=\frac{9}{5}\end{array}$

So, the slope of perpendicular line to our given line is $\frac{9}{5}$ .

Now we will substitute the slope and y -intercept $\left(0,3\right)$ in slope-intercept form of equation as:

  $\begin{array}{l}y=mx+b\left(\text{Slope-intercept form of equation}\right)\\ y=\frac{9}{5}\cdot x+3\left(\text{Substituting }m=\frac{9}{5}\text{ and }b=3\right)\\ y=\frac{9}{5}x+3\end{array}$

Therefore, the equation of perpendicular lineto our given line is $y=\frac{9}{5}x+3$ .