Math Equations for Parallel and Perpendicular Lines

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Perpendicular LinesIt’s fun and very instructive to figure out the math equations for parallel and perpendicular lines.

The basic mathematical equation for a line is,

ax + by = c

Here are three examples of line equations:

  • 2x + 3y = 6
  • 4x – 2y = –5
  • –x/3 + 2.47y = √3

Slope-Intercept Form

One of the most useful formats for the equation of a line is the slope-intercept form. That form is written,

y = mx + b

The variables here are x and y. The letters m and b are constants that represent the rise or tilt of the line (slope, m) and the point at which the line crosses the y-axis (intercept, b).

So the first of the three equations for a line listed above is written in the slope-intercept format as,

y = 2/3 x + 2

This tells us,

m = 2/3 and b = 2

Our desire now is to write equations for parallel and perpendicular lines based on the slope-intercept equation as cited above.

Deriving Equations for Parallel Lines

Two parallel lines have the same slope but a different intercept.

If the first line is written y = mx + b, then a parallel line may be written y = mx + b1.

Using our example line, y = 2/3 x + 2, all of the following are parallel lines:

  • y = 2/3 x + 8
  • y = 2/3 x – 7/16
  • y = 2/3 x + √13

For Perpendicular Lines

There are two differences for perpendicular lines. At first, you might be inclined to think as I once did, that you just take the inverse of the slope. This sounds logical, but is wrong. If the slope of the line is positive, it rises as x increases. The inverse of that slope is also positive and rises as x increases. But a perpendicular of such a line must rise to the left rather than to the right.

For that reason, the sign of the slope must also change. This leads us to the correct conclusion that a line with 2/3 slope is perpendicular to a line with a slope of  –3/2. That is,

y = –3/2 x + 2 [perpendicular line]

See the image.

There is an in-depth online proof of this located on the Central Oregon Community College website.

Note: You might also enjoy Point on a Line, a Line on a Plane, and a Plane in Space

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