Question

In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

line $y=-\frac{4}{5}x+2$ point (8, 9)

Short answer

Equation of the perpendicular line to$y=-\frac{4}{5}x+2$passing through the point (8, 9) is$y=\frac{5}{4}x-1$.

Step-by-step solution

Step 1. Given information

We have given equation of the line is,

$y=-\frac{4}{5}x+2$and the point is (8, 9).

Step 2. Concepts

We know that the slope of two perpendicular line is negative reciprocal of each other.

It means if slope of the original line is m then the slope of the perpendicular line is -1 / m.

We will use slope - point formula:

$y-y_1=m(x-x_1)$

Step 3. Explanation

We have given equation is,

$y=-\frac{4}{5}x+2$

Comparing it with slope intercept form we y = mx + b.

We get slope of the original line is,

$m=-\frac{4}{5}$

Hence, slope of the perpendicular line is$\frac{5}{4}$.

Using slope - point formula,

$y-y_1=m(x-x_1)$

Substituting the given point and slope,

$y-9=\frac{5}{4}(x-8)$

Simplifying it,

$y=\frac{5}{4}x-1$

Step 4. Conclusion

Equation of the perpendicular line to $y=-\frac{4}{5}x+2$passing through the point (8, 9) is, $y=\frac{5}{4}x-1$.