How to Find the Equation of a Parallel or Perpendicular Line

In linear algebra, if you are given the equation of a line and the coordinates of a point outside the line, you can find the equation of a parallel line that goes through the given point. You can also determine the equation of a perpendicular line that goes through the given point.

The standard equation form for a line in the xy-coordinate plane is y = mx + b, where m is the slope of the line (rise over run) and b is the y-intercept. A line that is horizontal has a slope of zero, and its equation is simply y = b. A line that is vertical has infinite slope, and its equation is x = a, where a is the x-intercept.

Follow the instructions below to compute parallel and perpendicular lines by hand, or use the calculator on the left to find the equations of the lines.

Parallel Lines

When two lines are parallel, their slopes are equal. That is, if the equation of the first line is y = mx + b, then the equation of a parallel line is y = mx + f, where f is a different y-intercept. When two lines are parallel, the only thing that is different between them is where they cross the y-axis. Here is an example of how to find the equation:

Suppose the equation of a line is y = 3x - 4 and a point has coordinates (10, 13). The equation of the parallel line will be y = 3x + f, where f depends on the coordinates of the point. It is easy to solve for f, just plug in 10 for x and 13 in for y. Then

13 = 3(10) + f
13 = 30 + f
-17 = f

So the equation of the parallel line is y = 3x - 17.

Perpendicular Lines

When two lines are perpendicular, their slopes are opposite reciprocals. That is, if the equation of the first line is y = mx + b, then the equation of a perpendicular line is y = -(1/m)x + g, where g is a different y-intercept. You can compute the equation of a perpendicular line using the same method above. Example:

Suppose the equation of a line is y = 3x - 4 and a point has coordinates (10, 13). The equation of the perpendicular line will be y = -(1/3)x + g, where g depends on the coordinates of the point. If we plug in 10 for x and 13 for y, then

13 = -(1/3)(10) + g
13 = -(10/3) + g
(49/3) = g, or
16.333 = g

So the equation of the perpendicular line is y = -(1/3)x + (49/3).

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