1
Equal slopes mean parallel, provided the lines are different
If the intercepts also agree the two equations describe one line, which is not usually what "parallel" is meant to include. Check b as well as m before answering.
Lesson · Lines, Functions and Graphs · Lines
Parallel and Perpendicular Lines
Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1. Both tests are stated for lines that have slopes, which leaves the vertical case to be handled separately and deliberately.
parallel: m₁ = m₂ (and different intercepts)
perpendicular: m₁m₂ = −1
so m₂ = −1/m₁ the NEGATIVE RECIPROCAL
x = a ⊥ y = b true, but not by the product rule
Two tests and one exception
Core ideas
What this lesson establishes
Both relationships are about direction alone, and slope is exactly the number that captures direction. That is why both tests are one line long.
2
Perpendicular means negative reciprocal
Flip the fraction and change the sign: 2/3 becomes −3/2. Doing only one of the two operations is the standard error, and each alone leaves a line that is not perpendicular to anything relevant.
3
The vertical case sits outside both tests
m₁m₂ = −1 presupposes two slopes. A vertical and a horizontal line are perpendicular by geometry, but no product can show it because one slope does not exist.
The material
Comparing two lines
One test for each relationship, then the case the tests exclude.
Definition · Parallel
Two distinct lines are parallel when they never meet. For non-vertical lines this happens exactly when their slopes are equal.
The connection to slope is easy to believe and easy to justify: equal slopes mean equal rise per unit run, so the vertical gap between the lines is the same at every x. If that gap is zero the lines coincide, and if it is anything else they never touch.
In slope-intercept form the test is pure inspection. y = 2x + 1 and y = 2x − 5 are parallel: same m, different b. This is what “Slope-Intercept Form” was buying — and note that 4x − 2y = 10 is the second of those lines in disguise, which no amount of staring at the standard form would reveal.
Definition · Perpendicular
Two lines are perpendicular when they meet at a right angle. For lines with slopes m₁ and m₂, this happens exactly when m₁m₂ = −1, equivalently m₂ = −1/m₁.
The negative and the reciprocal do separate jobs, and separating them is what makes the rule stick. The reciprocal exchanges rise and run — a quarter turn swaps the horizontal and vertical legs of the slope triangle — and the negative sign reverses direction, because a quarter turn sends a rising line to a falling one.
m1 perpendicular m2 product
------- -------------------- -------
2 -1/2 -1
-3 1/3 -1
3/4 -4/3 -1
-2/3 3/2 -1
1 -1 -1
Every line of that table is one calculation: flip, then negate. The last row is the one to remember as a picture — y = x and y = −x visibly cross at a right angle, and their slopes multiply to −1.
The vertical exception
A vertical line x = a and a horizontal line y = b are perpendicular; you can see the right angle. But the horizontal line has m = 0 and the vertical line has no slope, so the product test cannot be applied: there is no second number to multiply, and 0 times anything is 0, never −1.
This is not a flaw in the rule, it is the rule's stated scope. The theorem is about lines that have slopes. Handle vertical lines by recognising them first: perpendicular to a vertical line means horizontal, and perpendicular to a horizontal line means vertical, both settled without any arithmetic.
Example · Through (1, 2), parallel and perpendicular to y = −3x + 7
Parallel: same slope −3, so y − 2 = −3(x − 1) gives y = −3x + 5. Perpendicular: slope 1/3, so y − 2 = (1/3)(x − 1) gives y = (1/3)x + 5/3. Both pass through (1, 2), and −3 × 1/3 = −1.
Notice the shape of both answers: identify the required slope, then hand it to “Point-Slope and Standard Form”'s point-slope form. Every question of this type reduces to those two moves, and the only new content in this lesson is how to get the slope.
Interactive
Parallel and perpendicular
The product of the slopes, and when there is no product
Method
Finding a line through a point, parallel or perpendicular to a given line
Get the given slope, transform it, then it is “Point-Slope and Standard Form”.
1
Put the given line into slope-intercept form
Solve for y and read m. If there is no y the given line is vertical — jump to the exception and answer y = k or x = k directly.
2
Decide the slope you need
Parallel: the same m. Perpendicular: flip the fraction and change the sign. Do both operations, and say the new slope aloud before using it.
3
Use point-slope with the given point
y − y₁ = m(x − x₁). The point comes from the question and is not affected by which relationship was asked for.
4
Check the point and the relationship
Substitute the point into your final equation, then confirm the slopes are equal, or that their product is −1. Two checks, because there are two ways to get this wrong.
Slopes compared
Score 0 / 0Choose the answer you can justify.
Through (1, 2), parallel and perpendicular to y = −3x + 7
The given line is already in slope-intercept form, so m = −3 is there to be read.
PARALLEL required slope m = -3
y - 2 = -3(x - 1)
y - 2 = -3x + 3
y = -3x + 5
check (1, 2): -3(1) + 5 = 2 TRUE
check slopes: -3 = -3 parallel
PERPENDICULAR required slope m = -1/(-3) = 1/3
y - 2 = (1/3)(x - 1)
y - 2 = (1/3)x - 1/3
y = (1/3)x + 5/3
check (1, 2): 1/3 + 5/3 = 6/3 = 2 TRUE
check product: -3 · 1/3 = -1 perpendicular
the two answers are NOT parallel to each other, and both pass through (1, 2)
The constant in the perpendicular case comes out as 2 − 1/3 = 5/3. Fractions are the normal result here and rounding them to 1.67 throws away the exactness that made the check 6/3 = 2 conclusive.
Both checks are worth doing. The point check catches an arithmetic slip in the simplification; the slope check catches having flipped without negating, which the point check would happily pass.
For a faded pair, use the point (−2, 3) and the line 2x + 5y = 10. The supplied first move is y = −(2/5)x + 2, so the given slope is −2/5. Find the required parallel and perpendicular slopes, build both lines through the point, and perform both the point check and the relationship check.
Common mistakes
1
Flipping without negating, or negating without flippingPerpendicular to slope 3/4 is −4/3. 4/3 and −3/4 are each half of the operation, and neither line is perpendicular to the original.
2
Calling two identical lines parallely = 2x + 1 and 2y = 4x + 2 have the same slope because they are the same line. Compare intercepts after comparing slopes, and say coincident when that is what they are.
3
Forcing the product rule onto a vertical lineThere is no number to multiply. Recognise the vertical line first: its perpendiculars are exactly the horizontal lines, and the answer needs no arithmetic.
Check your understanding
✓
Finish when you can produce the parallel and the perpendicular line through a given point, and handle a vertical given line without reaching for the formula.
Given four lines — one in standard form, one vertical, one horizontal, one with a fractional slope — and a point, you should produce both companions for each and justify the vertical case in words. Knowing why the product test does not apply there is the difference between using a theorem and reciting it.
Note: Systems and Matrices solves pairs of linear equations. The three possible outcomes there — one solution, none, or infinitely many — are exactly the three cases here: different slopes, equal slopes with different intercepts, and the same line twice. The algebra in that course is counting what this lesson sees.