Equations of Parallel and Perpendicular Lines

Learning Outcomes

  • Write equations of parallel and perpendicular lines

Write the equations of parallel and perpendicular lines

The relationships between slopes of parallel and perpendicular lines can be used to write equations of parallel and perpendicular lines.

Let’s start with an example involving parallel lines.

Example

Write the equation of a line that is parallel to the line [latex]x–y=5[/latex] and goes through the point [latex](−2,1)[/latex].

Determine the Equation of a Line Parallel to Another Line Through a Given Point

Determine the Equation of a Line Perpendicular to Another Line Through a Given Point

When you are working with perpendicular lines, you will usually be given one of the lines and an additional point. Remember that two non-vertical lines are perpendicular if the slope of one is the negative reciprocal of the slope of the other. To find the slope of a perpendicular line, find the reciprocal, and then find the opposite of this reciprocal.  In other words, flip it and change the sign.

Example

Write the equation of a line that contains the point [latex](1,5)[/latex] and is perpendicular to the line [latex]y=2x– 6[/latex].

Determine the Equation of a Line Perpendicular to a Line in Slope-Intercept Form

Example

Write the equation of a line that is parallel to the line [latex]y=4[/latex] through the point [latex](0,10)[/latex].

Example

Write the equation of a line that is perpendicular to the line [latex]y=-3[/latex] through the point [latex](-2,5)[/latex].

Try It

Find the Equation of a Perpendicular and Horizontal Line to a Horizontal Line

Summary

When lines in a plane are parallel (that is, they never cross), they have the same slope. When lines are perpendicular (that is, they cross at a [latex]90°[/latex] angle), their slopes are opposite reciprocals of each other. The product of their slopes will be [latex]-1[/latex], except in the case where one of the lines is vertical causing its slope to be undefined. You can use these relationships to find an equation of a line that goes through a particular point and is parallel or perpendicular to another line.

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