Linear Functions
Parallel and Perpendicular Lines
Equal slopes for parallel, negative reciprocals for perpendicular, why the second rule holds, and the vertical-horizontal exception.
What this page covers
- Test two lines for parallelism from their equations
- Test two lines for perpendicularity
- Find a line through a given point parallel or perpendicular to another
- Handle the vertical and horizontal exception
Parallel lines
Two distinct lines are parallel exactly when their slopes are equal. The source states it plainly: parallel lines have the same slope, or gradient.
It follows from the definition of slope as a rate. Two lines climbing at the same rate never converge, and two climbing at different rates must eventually meet.
| Equation | Slope | y-intercept |
|---|---|---|
| y = 2x - 3 | 2 | -3 |
| y = 2x + 1 | 2 | 1 |
| y = 2x | 2 | 0 |
| y = 2x - 4 | 2 | -4 |
All four have slope 2 and differ only in where they cross the y-axis. The source draws them as a family of parallel lines, which is exactly what varying b while fixing m produces.
Equal slopes and equal intercepts means the same line, not two parallel ones. Lines are usually called parallel only when distinct.
Worked example — the source's case
Find the line through (-3, 5) parallel to y = 5x - 4.
The slope must be 5. Using point-slope form:
At x = -3: y = -15 + 20 = 5 ✓. The slope 5 matches the original, and the intercept 20 differs from -4, so the lines are distinct.
Perpendicular lines
The second slope is the negative reciprocal of the first: invert the fraction and reverse the sign.
The source illustrates with a slope of 23 and its perpendicular -32, drawing the rise and run swapping roles as the line turns through a right angle.
| Slope | Perpendicular slope | Product |
|---|---|---|
| 23 | -32 | -1 |
| 3 | -13 | -1 |
| -13 | 3 | -1 |
| -2 | 12 | -1 |
| 1 | -1 | -1 |
The source's own example is a line of slope 3 against one of slope -13, checking 3 × -13 = -1.
Why the product is -1
Take a line of slope pq, so travelling q right raises it by p. Rotating the whole picture through 90° turns the horizontal step into a vertical one and the vertical step into a horizontal one, with one of them reversing direction.
The rotated line therefore travels -p horizontally while rising q, giving slope q-p = -qp. Multiplying:
The same statement appears in vector language: two lines are perpendicular when their direction vectors have zero dot product. For directions (q, p) and (-p, q) the dot product is -qp + pq = 0, which is the same fact. See The Dot Product.
The exception
A vertical line and a horizontal line are perpendicular, but the vertical one has undefined slope, so no product can be formed. The rule m1m2 = -1 does not apply and is not needed — the perpendicularity is obvious.
| Line 1 | Line 2 | Parallel? | Perpendicular? |
|---|---|---|---|
| y = mx + b1 | y = mx + b2 | Yes, if b1 ≠ b2 | Only if m2 = -1, impossible |
| y = mx + b | y = -1mx + c | No | Yes |
| x = a | x = c | Yes, if a ≠ c | No |
| y = k | y = c | Yes, if k ≠ c | No |
| x = a | y = k | No | Yes — but the slope rule does not apply |
Worked example: a perpendicular through a point
Perpendicular to 2x + 3y = 12 through (1, 4)
- Find the original slope. From Ax + By = C the slope is -AB = -23.
- Take the negative reciprocal. m = 32.
- Use point-slope form with the given point.
- Convert to whatever form is wanted.
At (1, 4): 3 - 8 = -5 ✓. Slopes: -23 × 32 = -1 ✓.
Combining this with the midpoint formula gives the perpendicular bisector of a segment: find the midpoint, find the segment's slope, take the negative reciprocal, and use point-slope form.
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| Reciprocal without the sign change | Negative reciprocal | The product must be -1 |
| Sign change without the reciprocal | Both are needed | 23 → -23 gives product -49 |
| Reading the slope of Ax + By = C as AB | It is -AB | Rearrange to check |
| Applying the rule to a vertical line | No slope exists | Perpendicularity is obvious there |
| Calling identical lines parallel | Parallel lines are distinct | Compare intercepts too |
| Using the wrong point in point-slope form | Use the point the new line must pass through | Substitute it back |
Frequently asked questions
Why do perpendicular slopes multiply to -1?
Rotating a line through 90° swaps rise and run and reverses one sign. A slope of 23 becomes -32, and the product is -1.
Does the rule ever fail?
Yes, for a vertical and a horizontal line. They are perpendicular, but one has undefined slope so no product can be formed. It is the only exception.
How do I find a perpendicular slope quickly?
Turn the fraction upside down and change its sign. From 34 to -43; from -2 = -21 to 12.
Do parallel lines have the same intercept?
No — if they did they would be the same line. Parallel lines have equal slopes and different intercepts. The source draws four lines of slope 2 with intercepts -3, +1, 0 and -4.
Source. Handwritten teaching notes, Week 5, pages 9-10.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.