The slopes of two lines tell you straight away whether they are parallel, perpendicular or neither.
Parallel lines: the same slope
Parallel lines never meet, so they must slant by exactly the same amount. Two lines are parallel when their slopes are equal and their y-intercepts are different.
- y = 2x + 1 and y = 2x − 3 are parallel: both slopes are 2.
- y = ½x + 1 and y = ½x − 2 are parallel: both slopes are ½.
Perpendicular lines: negative reciprocals
Perpendicular lines meet at a right angle. Their slopes are negative reciprocals: flip the fraction and change the sign. Another way to say this is that the two slopes multiply to −1.
- Slope 2 and slope −1/2: 2 × (−1/2) = −1
- Slope 3/4 and slope −4/3: (3/4) × (−4/3) = −1
- Slope −5 and slope 1/5: (−5) × (1/5) = −1
The exception is a horizontal line (slope 0) and a vertical line (undefined slope). They are perpendicular, but you can’t multiply their slopes.
Worked examples
Example 1: a parallel line through a point
Find the line parallel to y = 3x − 2 that passes through (1, 4).
A parallel line has the same slope, 3. Substitute the point into y = 3x + b: 4 = 3 × 1 + b, so b = 1.
The answer is y = 3x + 1.
Example 2: a perpendicular line through a point
Find the line perpendicular to y = ⅖x + 1 that passes through (2, −3).
The perpendicular slope is the negative reciprocal of 2/5, which is −5/2. Substitute: −3 = −5/2 × 2 + b, so −3 = −5 + b, and b = 2.
The answer is y = −5/2 x + 2.
Example 3: checking two lines
Are 2x + 3y = 6 and 3x − 2y = 4 parallel, perpendicular or neither?
Rearranged, the first is y = −⅔x + 2, with slope −2/3. The second is y = 3/2 x − 2, with slope 3/2.
(−2/3) × (3/2) = −1, so the lines are perpendicular.
A tip for graphs
Perpendicular lines only look like they meet at a right angle when both axes use the same scale. If the x-axis counts in 1s and the y-axis counts in 5s, the right angle looks squashed. Trust the slopes rather than the picture.