# How to Find the Equation of a Parallel or Perpendicular Line

Parallel and Perpendicular Lines
Linear Equation Y = X +
Point Coordinates ( , )
Parallel Line:

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).