## 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.

### Example

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

## 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 $(1,5)$ and is perpendicular to the line $y=2x– 6$.

## 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 $y=4$ through the point $(0,10)$.

### Example

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

## 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 $90°$ angle), their slopes are opposite reciprocals of each other. The product of their slopes will be $-1$, 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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