Linear Functions

Parallel & Perpendicular Lines

Writing equations of lines parallel or perpendicular to a given line.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Point-Slope Form

What Slope Reveals About How Two Lines Relate

Two lines never cross, no matter how far they extend. Two other lines cross at a perfect right angle. Before reading on, think about what must be true of each pair's slopes — not their y-intercepts, just their slopes — for these two very different relationships to hold.

Definition — Parallel and Perpendicular Slopes

Two distinct lines are parallel exactly when they have the same slope. Two lines are perpendicular exactly when their slopes are negative reciprocals of each other — their product equals −1.

Parallel lines must share a slope because slope is a rate of change — if two lines climbed or fell at different rates, they'd inevitably drift together or apart no matter where they started, eventually crossing. Only an identical rate of change keeps two distinct lines the same distance apart forever.

Perpendicularity is a little more surprising. Picture a line's rise-over-run direction, like 3/1. Rotating that direction a quarter turn swaps the roles of rise and run and reverses one of the signs — turning 3/1 into −1/3. That swap-and-negate pattern is exactly what 'negative reciprocal' means, and it's why perpendicular slopes always multiply to exactly −1: m × (−1/m) = −1.

Worked Example — Writing a Parallel Line's Equation

Write the equation of the line through (4, 1) that's parallel to y = 2x − 5. A parallel line shares the same slope, 2. Using point-slope form with (4, 1): y − 1 = 2(x − 4), which simplifies to y = 2x − 7.

Worked Example — Writing a Perpendicular Line's Equation

Write the equation of the line through (6, 2) that's perpendicular to y = (1/3)x + 4. The given slope is 1/3, so the perpendicular slope is the negative reciprocal: −3. Using point-slope form: y − 2 = −3(x − 6), which simplifies to y = −3x + 20.

Worked Example — Verifying Perpendicularity

Are the lines y = 4x + 1 and y = −(1/4)x + 9 perpendicular? Multiply their slopes: 4 × (−1/4) = −1. Since the product is exactly −1, the lines are perpendicular.

Graph Visualizer

Trace modePoint mode (click graph to plot)

Show x^2

Color

Expression

Domain & range

x min

x max

y min

y max

Precision (decimal places)2

Evaluate a point

Evaluate at x =

x^2 = 0

Tip

To find a perpendicular slope quickly, flip the original slope's fraction and switch its sign — a slope of 2 (or 2/1) becomes −1/2; a slope of −3/5 becomes 5/3.

Common Mistakes

Using the same slope for a perpendicular line instead of the negative reciprocal.
Parallel lines share the exact same slope; perpendicular lines need the negative reciprocal — flip the fraction and change the sign, don't just copy the original slope.
Forgetting to change the sign when finding a negative reciprocal, such as finding 1/4 as the perpendicular slope for 4 instead of −1/4.
A negative reciprocal requires two changes together: flipping the fraction and reversing the sign — both steps are needed, not just one.

Key Takeaways

Parallel lines have identical slopes, since they must maintain the same rate of change to never converge.
Perpendicular lines have slopes that are negative reciprocals of each other, multiplying to exactly −1.
Point-slope form makes it straightforward to write a parallel or perpendicular line's equation once its slope is determined.

Summary

Slope alone reveals whether two lines are parallel, perpendicular, or neither. With linear equations and their graphs fully connected, the next unit asks a new question: what happens where two different lines' graphs meet.

Track your progress

← Point-Slope Form Back to Linear Functions