Part 13 of 20 · LinesSee the whole course
Parallel and perpendicular lines: the slope rules
Parallel lines have the same slope. Perpendicular lines have slopes that multiply to −1: flip the fraction and change the sign. The slope of a line perpendicular to slope 2 is −½.
Parallel lines: the same slope
Two lines that never meet climb at exactly the same rate. So they have the same slope m and different intercepts b. The lines y = 2x + 1 and y = 2x − 3 are parallel: both have m = 2.
If the intercepts are also the same, the two equations describe the same line, not two parallel ones.
Perpendicular lines: slopes that multiply to −1
Perpendicular lines meet at a right angle, like the corner of a page. Their slopes are negative reciprocals: do two things to the slope.
- Flip the fraction. 2 is 2/1, so it becomes 1/2.
- Change the sign. Positive becomes negative, so it becomes −1/2.
Check by multiplying: 2 × (−½) = −1. Whenever two slopes multiply to −1, the lines are perpendicular. Slide the orange line below and watch for the little square that marks a right angle. It appears only at slope −½.
Write the equation of a parallel line
Find the line parallel to y = 3x − 1 that passes through (1, 5).
Same slope: m = 3, so y = 3x + b
Put in the point (1, 5): 5 = 3(1) + b
b = 2
So the line is y = 3x + 2
The slope comes from the rule. The intercept comes from the point. This is the same method as the equation of a line from two points.
Write the equation of a perpendicular line
Find the line perpendicular to y = 2x + 1 that passes through (4, 1).
Perpendicular slope: flip 2 and change the sign, so m = −1/2
Put in the point (4, 1): 1 = −½(4) + b
1 = −2 + b, so b = 3
So the line is y = −½x + 3
Watch it explained in under a minute
Special case: horizontal and vertical lines
A horizontal line (slope 0) and a vertical line (slope undefined) are perpendicular, even though the multiplying rule does not apply to them. See horizontal and vertical lines for why.
The four mistakes to avoid
- Flipping the fraction but keeping the sign. The perpendicular slope to 2 is −1/2, not 1/2.
- Changing the sign but not flipping. Slopes 2 and −2 multiply to −4, so those lines are not perpendicular.
- Keeping the old intercept. A new line through a new point needs its own b, so use the point to find it.
- Calling lines parallel because they look close. Compare the slopes: they must be exactly equal.
Try it yourself
Tap a question to see the answer.
1. Are y = 3x + 1 and y = 3x − 5 parallel?
Both slopes are 3, and the intercepts differ, so they are parallel.
2. What is the slope of a line perpendicular to y = 4x − 2? And to y = −⅔x + 1?
For slope 4: flip to 1/4 and change the sign, so −1/4 (check: 4 × (−¼) = −1). For slope −2/3: flip to 3/2 and change the sign, so 3/2 (check: −⅔ × 3/2 = −1).
3. Find the line through (3, 2) that is perpendicular to y = −⅓x + 5.
The perpendicular slope is 3 (flip −1/3 and change the sign). Put in (3, 2): 2 = 3(3) + b, so b = −7. The line is y = 3x − 7.
Where this fits in
Parallel and perpendicular lines finish the course on lines. From here we move on to equations, starting with how to solve a linear equation.