How to Find Perpendicular Slope

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Knowing two points on a line, (​x1, ​y1) and (​x2, ​y2), allows you to calculate the slope of the line (​m​), because it's the ratio ∆​y​/∆​x​:

m = \frac{y_2 - y_1}{x_2 - x_1}

If the line intersects the y-axis at b, making one of the points (0, ​b​), the definition of slope produces the slope intercept form of the line ​y​ = ​mx​ + ​b​. When the equation of the line is in this form, you can read slope directly from it, and that allows you to immediately determine the slope of a line perpendicular to it because it's the negative reciprocal.

TL;DR (Too Long; Didn't Read)

The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. If the given line has slope ​m​, the slope of a perpendicular line is −1/m.

Procedure for Determining Perpendicular Slope

By definition, the slope of the perpendicular line is the negative reciprocal of the slope of the original line. As long as you can convert a linear equation to slope intercept form, you can easily determine the slope of the line, and since the slope of a perpendicular line is the negative reciprocal, you can determine that as well.

Your equation may have ​x​ and ​y​ terms on both sides of the equals sign. Collect them on the left side of the equation and leave all the constant terms on the right side. The equation should have the form

Ax + By = C

where ​A​, ​B​ and ​C​ are constants.

The form of the equation is ​Ax​ + ​By​ = ​C​, so subtract ​Ax​ from both sides and divide both sides by ​B​. You get :

y = -\frac{A}{B}\,x +\frac{C}{B}

This is the slope intercept form. The slope of the line is −(​A​/B).

The slope of the line is −(​A​/​B​), so the negative reciprocal is ​B​/​A​. If you know the equation of the line in standard form, you simply need to divide the coefficient of the y term by the coefficient of the ​x​ term to find the slope of a perpendicular line.

Keep in mind that there are an infinite number of lines with slope perpendicular to a given line. If you want the equation of a particular one, you need to know the coordinates of at least one point on the line.

Examples

1. ​What is the slope of a line perpendicular to the line defined by

3x + 2y = 15y - 32

To convert this equation to standard from, subtract 15y from both sides:

3x + (2y - 15y) = (15y - 15y) - 32

After performing the subtraction, you get

3x -13y = -32

This equation has the form ​Ax​ + ​By​ = ​C​. The slope of a perpendicular line is ​B​/​A​ = −13/3.

2. What is the equation of the line perpendicular to 5​x​ + 7​y​ = 4 and passing through the point (2,4)?

Start be converting the equation to slope intercept form:

y = mx + b

To do this, subtract 5​x​ from both sides and divide both sides by 7:

y = -\frac{5}{7}x + \frac{4}{7}

The slope of this line is −5/7, so the slope of a perpendicular line must be 7/5.

Now use the point you know to find the ​y​-intercept, ​b​. Since ​y​ = 4 when ​x​ = 2, you get

4 = \frac{7}{5} × 2 + b \\ \,\\ 4 = \frac{14}{5} + b \text{ or } \frac{20}{5} = \frac{14}{5} + b \\ \,\\ b = \frac{20 - 14}{5} = \frac{6}{5}

The equation of the line is then

y = \frac{7}{5} x + \frac{6}{5}

Simplify by multiplying both sides by 5, collect the x and y terms on the right side and and you get:

-7x + 5y = 6
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