Parallel & Perpendicular Lines: Slope Rules

TL;DR

Parallel and perpendicular lines are two relationships you can read straight off slopes: parallel lines have equal slopes (m1=m2), and perpendicular lines have slopes that multiply to −1 (m1⋅m2=−1). This article covers both slope rules, why each works, how to tell lines apart from their equations, the special vertical-line case, and six worked examples.

What Are Parallel and Perpendicular Lines?

Parallel lines are two lines in the same plane that never meet, no matter how far they run. They keep a constant gap because they rise at the same rate. Perpendicular lines are two lines that cross at a right angle, exactly 90∘. Both relationships are fixed by a single feature of each line: its slope, the change in y for each unit of change in x.

Parallel: the slopes are equal, m1=m2.
Perpendicular: the slopes are negative reciprocals, so their product is −1: m1⋅m2=−1.

The Parallel Slope Rule: Equal Slopes

Two non-vertical lines are parallel exactly when their slopes are equal: m1=m2.

The reasoning is direct. Slope is the rate a line climbs as it moves right. If two lines climb at the same rate from start to finish, they hold the same direction the whole way, so the vertical gap between them never changes, and lines that never close the gap never meet. A different slope, even slightly, means one line gains on the other and they eventually cross.

One guard rail: equal slopes alone are not quite enough. If two lines have the same slope and the same y-intercept, they are not parallel; they are the same line lying on top of itself. True parallel lines need equal slopes and different y-intercepts.

y=3x+1 and y=3x−4 are parallel: same slope 3, different intercepts.
y=3x+1 and y=3x+1 are the same line, not parallel.

The Perpendicular Slope Rule: Negative Reciprocals

Two non-vertical lines are perpendicular exactly when the product of their slopes is −1:

m1⋅m2=−1.

Rearranged, this says one slope is the negative reciprocal of the other: m2=−1/m1. To build a perpendicular slope from a given one, do two things, flip it and switch its sign:

Start with slope 2, written 2/1. Flip to 1/2, switch the sign to −1/2. Check: 2⋅(−1/2)=−1.
Start with slope −3/4, flip to −4/3, switch the sign to 4/3. Check: −3/4⋅4/3=−1.

How Do You Tell If Two Lines Are Parallel or Perpendicular From Their Equations?

This is the question that shows up most on homework, and the method is the same every time, get each line into slope-intercept form y=mx+b, read off the slope, then compare.

Solve each equation for y so it reads y=mx+b.
Read the slope m from each (the coefficient of x).
Compare the slopes:
- Equal slopes, different intercepts → parallel.
- Product of slopes =−1 → perpendicular.
- Neither → the lines just cross at some non-right angle.

For example, 2x+y=5 rearranges to y=−2x+5 (slope −2), and x−2y=6 rearranges to y=1/2x−3 (slope 1/2). Product: −2⋅1/2=−1, so the lines are perpendicular.

What About Vertical and Horizontal Lines?

The two slope rules cover every case except one pair. A horizontal line has slope 0; a vertical line has an undefined slope. A horizontal and a vertical line are clearly perpendicular, they meet at a right angle, yet you cannot check that with the product rule, because 0×(undefined) is not −1 or anything else.

So treat this pair as a known exception: a horizontal line and a vertical line are always perpendicular, even though the negative-reciprocal test does not apply. Two vertical lines, meanwhile, are always parallel.

Examples of Parallel and Perpendicular Lines

With both slope rules and the equation method in place, here is the concept doing real work.

Example 1 - Are y=4x+2 and y=4x−7 parallel, perpendicular, or neither?

Final answer: parallel.

Example 2 - Are the lines y=2x+1 and y=−2x+5 perpendicular?

Final answer: neither.

Example 3 - Find the slope of a line perpendicular to y=−3/5x+2. Final answer: 5/3.

Example 4 - A line is parallel to y=−2x+9 and passes through (0,3). Write its equation. Final answer: y=−2x+3.

Example 5 - Are 3x+y=6 and x−3y=9 parallel, perpendicular, or neither? Final answer: perpendicular.

Example 6 - A vertical line is x=4 and a horizontal line is y=−1. Are they perpendicular? Final answer: perpendicular.

Key Takeaways

Parallel lines have equal slopes and different y-intercepts.
Perpendicular lines have slopes that are negative reciprocals.
To compare lines from equations, rearrange each to y=mx+b, then read and compare the slopes.
A horizontal line and a vertical line are always perpendicular, the one pair the product rule cannot test.