Quarry School

Compare parallel and perpendicular lines

Explain it like I am five

Picture two straight train tracks and two edges of a square tile. The tracks keep the same direction and never meet. Parallel lines do that. The tile's edges meet at a square corner, called a right angle, which measures 90 degrees, written 90°. Perpendicular lines do that. To compare equations, look at their slopes. Equal slopes keep the same slant. A perpendicular slope comes from turning the slant through a quarter turn, so the up-and-across movements trade roles and one direction changes sign. If two equations have both the same slope and the same starting height, they describe one line twice. Those are coincident lines, meaning the very same line.

−5−4−3−2−1123456−6−4−22468(0, −4)(8, 0)
The solid g line and dashed h line have negative reciprocal slopes.
Reminder
  • Isolating y. 6x − 2y = 8 becomes y = 3x − 4 by subtracting 6x and dividing every term by −2.
  • Fraction multiplication. 12 × (−2) = −1 because −22 = −1.
  • Slope and intercept. In y = 2x − 6, m = 2 and b = −6.
  • Vertical lines. x = 7 has no slope number and must be handled separately.
  • Reciprocals. 25 × 52 = 1; reversing one sign makes the product −1.
Why it works. For parallel nonvertical lines, subtracting outputs gives (mx + b1) − (mx + b2) = b1 − b2. Their vertical separation stays constant and nonzero, so they cannot meet. For perpendicular lines, a right-angle turn changes a move right r and up s into a move left s and up r. The new slope is r−s, the negative reciprocal of sr. This argument needs nonzero finite slopes. A horizontal and a vertical line still make a right angle, but the vertical slope is undefined.
RuleRule: Nonvertical lines are parallel when m1 = m2 and b1 ≠ b2; coincident when both match. Lines with finite nonzero slopes are perpendicular when m1m2 = −1. Horizontal and vertical lines are perpendicular.
The same idea, five ways
Say it

Say: the same direction and different lines means parallel; a square corner means perpendicular; the same drawing twice means coincident.

Write it

Slopes identify the relationship once vertical-line exceptions are handled.

In math
  • parallel: m1 = m2, b1 ≠ b2
  • coincident: m1 = m2, b1 = b2
  • perpendicular: m2 = −1m1
  • m1m2 = −1
Like

Rail tracks keep a gap; square-tile edges make a right angle.

See it
−4−2246−6−4−22468(0, 3)(−1.5, 0)
These equal-slope lines have different starting heights, so they never meet.
The same idea, other ways
As a fixed gap

For y = x + 1 and y = x − 2, the first output is always 3 larger. Matching slopes make that gap stay fixed.

−4−2246−6−4−22468(0, 1)(−1, 0)
The lines have the same rise for each run and stay separated.
As a quarter turn

A move right 2, up 1 has slope 12. Turn it through a right angle to get left 1, up 2, with slope −2. The product 12 × (−2) is −1.

right angle
A quarter turn exchanges the across and upright directions.
As a numerical check

Slopes 3 and −13 give product −1. Slopes 3 and −3 give product −9, so opposite signs alone are not enough.

3 × (−13) = −1: perpendicular
3 × (−3) = −9: neither parallel nor perpendicular
The product checks more than the signs.
.1Parallel lines

Parallel lines keep the same direction and never meet. For nonvertical lines, the starting heights must differ as well as the slopes matching. Distinct vertical lines are also parallel because their fixed across positions differ.

  • Rule: y = mx + b1 and y = mx + b2 are parallel if b1 ≠ b2.
  • Rule: x = a and x = c are parallel if a ≠ c.
−4−2246−6−4−22468(0, 1)(−1, 0)
Compare the two lines' slopes and intercepts.
Reminder
  • Slope-intercept form. In y = 3x − 4, the slope is 3 and the intercept is −4.
The same idea, five ways
Say it

Say: same direction, different lines.

Write it

Parallel lines do not share any point.

In math
  • m1 = m2, b1 ≠ b2
  • vertical case: x = a and x = c, a ≠ c
Like

Two straight rails never join.

See it
−4−2246−6−4−22468(0, 1)(−1, 0)
Equal slants with different intercepts make distinct parallel lines.
Worked exampleEqual slopes with a real gap

You need to decide whether y = x + 1 and y = x − 2 are parallel.

−4−2246−6−4−22468(0, 1)(−1, 0)
Compare the two lines' slopes and intercepts.
  1. Both slopes are 1, and their intercepts are 1 and −2.Their directions match but their starting heights differ.
  2. Their output difference is (x + 1) − (x − 2) = 3.A nonzero gap at every input means no shared point.
Answer
They are parallel.
Check At x = 0 the gap is 3; at x = 2 the outputs 3 and 0 still differ by 3.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: y = x + 1 and y = 2x + 1 are parallel because they start together.
The slopes differ, and they share (0, 1), so they meet.
✓ Instead: Their equal intercepts do not make them parallel. Compare slopes first.
Tips and tricks
  • Tip: Parallel slopes match, then check that the intercepts differ.
.2Coincident lines

Coincident lines are one line described twice, like two transparent copies laid exactly on top of each other. Every point of one is on the other. They are not two separated tracks.

  • Rule: Equal slopes and equal intercepts give coincident lines.
  • Rule: Different-looking equations can describe the same line after simplification.
−4−2246−6−4−22468(0, 3)(−1.5, 0)
Compare the two lines' slopes and intercepts.
Reminder
  • Division on both sides. 2y = 4x + 6 becomes y = 2x + 3 after every term divides by 2.
The same idea, five ways
Say it

Say: the same line twice.

Write it

Coincident lines share all their points.

In math
  • m1 = m2, b1 = b2
  • y = x + 1 and 2y = 2x + 2
Like

Two identical tracings overlap completely.

See it
−4−2246−6−4−22468(0, 1)(−1, 0)
The dashed copy lies on the solid line because both equations are identical after simplification.
Worked exampleDifferent writing, same line

You need to compare y = 2x + 3 and 2y = 4x + 6.

−4−2246−6−4−22468(0, 3)(−1.5, 0)
Compare the two lines' slopes and intercepts.
  1. Divide every term of the second equation by 2: y = 2x + 3.Balanced division reveals its slope and intercept.
  2. Both slopes are 2 and both intercepts are 3.They have the same start and the same movement.
Answer
The lines are coincident.
Check At any input x, doubling y = 2x + 3 gives 2y = 4x + 6, so each equation produces the other.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: equal slopes always mean two distinct parallel lines.
If the intercepts also match, there is no gap: the equations describe the same line.
✓ Instead: Check intercepts to distinguish parallel from coincident.
Tips and tricks
  • Tip: Memory cue: coincide means occupy the same place.
.3Perpendicular lines and negative reciprocals

A negative reciprocal means the flip with its sign reversed. A reciprocal alone is the number that multiplies the old number to make 1. Changing its sign makes the product −1. That is the slope needed for a right-angle meeting when neither slope is zero or undefined.

  • Rule: For nonzero m, the negative reciprocal is −1m.
  • Rule: 25 becomes −52; 3 becomes −13; −16 becomes 6.
  • Rule: A horizontal line and a vertical line are perpendicular without a slope-product calculation.
−4−224−4−224(0, 0)(−0, 0)
Slopes one half and negative two make a right angle.
Reminder
  • Reciprocals. 25 × 52 = 1 because numerator and denominator both equal 10.
  • Fraction signs. The negative reciprocal of a negative slope is positive: −16 becomes 6.
The same idea, five ways
Say it

Say: flip a nonzero slope, then reverse its sign.

Write it

Perpendicular nonvertical lines have slopes whose product is −1.

In math
  • m2 = −1m1
  • pq becomes −qp
  • m1m2 = −1
Like

Turning a ramp a quarter turn exchanges rise and run and reverses one direction.

See it
−4−224−4−224(0, 0)(−0, 0)
These lines with slopes 1 and −1 meet at a right angle.
Worked exampleFlip and reverse the sign

You need the perpendicular slope when the original slope is 25, and when it is −16.

25 ↔ −52
−16 ↔ 6
Each pair multiplies to −1
Both flipping and changing sign are required.
  1. For 25, flip to 52, then change the sign to −52.The flip makes a product of 1; changing its sign makes −1.
  2. Check: 25 × (−52) = −1010 = −1.The product verifies the perpendicular relationship.
  3. For −16, flip to −6, then change its sign to 6. Check (−16) × 6 = −1.A negative slope has a positive negative reciprocal.
Answer
  • For 25: −52.
  • For −16: 6.
Check Applying flip-and-sign again restores each original slope, so the relationship works both directions.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: the perpendicular slope to 3 is −3.
The product is −9, not −1. The sign changed but the reciprocal was missed.
✓ Instead: Write 3 = 31, flip to 13, then change sign to −13.
Tips and tricks
  • Tip: Check the product before accepting a perpendicular slope.
  • Tip: A picture can distort a right angle if one horizontal unit and one vertical unit have different screen lengths. Use the slope test or equal axis scales.
Strategy: step by step
  1. 1. Rewrite each nonvertical equation with y alone so its slope is visible.
  2. 2. Equal slopes and different intercepts mean parallel lines.
  3. 3. Equal slopes and equal intercepts mean coincident lines.
  4. 4. Otherwise multiply the slopes. A product of −1 means perpendicular lines.
  5. 5. Handle a vertical line separately: it is perpendicular to a horizontal line, and parallel to another distinct vertical line.
Strategy
Strategy: classify two lines
1
Is either line vertical?
YesIf both equations fix the same x, they are coincident; different fixed x-values mean parallel. If only one is vertical, a horizontal other line is perpendicular and any other line is neither.
NoCompare the slopes.
↓
2
Are the slopes equal?
YesDifferent intercepts mean parallel; equal intercepts mean coincident.
NoMultiply them.
↓
3
Is the product −1?
YesThey are perpendicular.
NoThey are neither parallel nor perpendicular.
  1. Isolate y in each nonvertical equation.
  2. Check equal slopes and then intercepts.
  3. For unequal finite nonzero slopes, multiply and compare with −1.
  4. Check horizontal/vertical pairs separately.
Worked exampleClassify the four verified functions

You need to find which pair of these lines never meets and which pair meets at a right angle: f(x) = 2x + 3, g(x) = 12x − 4, h(x) = −2x + 2, j(x) = 2x − 6.

−4−2246−6−4−22468(0, 3)(−1.5, 0)
Compare the two lines' slopes and intercepts.
−4−2246−6−4−22468(0, −4)(8, 0)
The g and h slopes multiply to −1.
  1. Read slopes: f has 2, g has 12, h has −2, and j has 2.Each slope is the coefficient of x in slope-intercept form.
  2. f and j have equal slopes 2 but different intercepts 3 and −6.Equal direction and a nonzero starting gap identify parallel lines.
  3. Multiply the slopes of g and h: 12 × (−2) = −1.A product of −1 identifies perpendicular nonvertical lines.
  4. Other unequal-slope products are 1 or −4, so no other pair is perpendicular; only f and j share slopes.Checking all relationships avoids choosing lines only by appearance.
Answer
  • Parallel: f and j.
  • Perpendicular: g and h.
Check The difference f(x) − j(x) = 9 stays nonzero. The negative reciprocal of 12 is −2, confirming the second pair by another route.
Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.
Rung 1Rung 1: matching slopes and different starts

You need to decide whether y = 5x + 8 and y = 5x − 3 are parallel, perpendicular, coincident, or neither.

−22−12−8−44812162024(0, 8)(−1.6, 0)first startsecond start
The equal-slope lines begin at different heights.
  1. Both coefficients of x are 5, so the slopes match.The equations already have y alone, making the slope numbers visible.
  2. The intercepts are 8 and −3, which differ. The lines are parallel.Equal slopes give the same direction; different starting heights make them distinct.
  3. (5x + 8) − (5x − 3) = 8 + 3 = 11.The x terms cancel, leaving a nonzero gap at every input.
Answer
  • Parallel.
  • Both slopes are 5; intercepts are 8 and −3.
Check At x = 0 the outputs are 8 and −3. At x = 1 they are 13 and 2. Both gaps are 11, agreeing with the constant-gap calculation.
Rung 2Rung 2: different-looking equations of one line

You need to compare y = −47x + 6 with 7y = −4x + 42.

−224682468(0, 6)(10.5, 0)sample shared point
The dashed and solid graphs overlap because the rewritten equations match.
  1. Divide every term of the second equation by 7: y = −47x + 427.Equal division preserves the same line points and isolates y.
  2. 427 = 6, so the second equation becomes y = −47x + 6.Seven groups of 6 total 42, so the constants match too.
  3. The slopes and intercepts both match. These equations are coincident.Matching direction and placement describe the same line, rather than separated parallel lines.
Answer
Coincident: the equations describe the same line.
Check At x = 7, the first gives y = −4 + 6 = 2. The original second equation gives 7 × 2 = −4 × 7 + 42 = 14. Multiplying the first formula by 7 reproduces the entire second equation.
Rung 3Rung 3: a fraction pair needs both tests

You need to decide whether y = 38x − 5 and y = −83x + 11 are perpendicular.

−10−8−6−4−22468101214−10−8−6−4−22468101214(0, −5)(13.3, 0)
The two slopes are negative reciprocals, which supplies the right-angle test.
  1. The slopes are 38 and −83, and they are unequal.Unequal slopes rule out parallel and coincident lines.
  2. Multiply: 38 × (−83) = −2424 = −1.The top products and bottom products both equal 24; the signs differ.
  3. The lines are perpendicular.Finite nonzero slopes with product −1 make a right-angle meeting.
Answer
Perpendicular.
Check Flipping 38 gives 83, and changing its sign gives −83, exactly the second slope. Opposite signs alone would not have proved the result.
Rung 4Rung 4: isolate y and handle an undefined slope

You need two classifications. First compare 8x + 3y = 29 with y = 38x − 17. Then compare x = 9 with y = −8.

−12−9−6−33691215−20−16−12−8−448121620(0, 9.67)(3.62, 0)
The isolated first slope is negative eight thirds; its product with three eighths is −1.
x = 9: vertical
y = −8: horizontal
Meeting point: (9, −8)
Perpendicular by directions
No undefined slope is put into a product.
  1. Subtract 8x from both sides of 8x + 3y = 29: 3y = 29 − 8x.This removes the x term from beside y so the slope can be exposed.
  2. Divide every term by 3: y = 293 − 83x = −83x + 293.Division leaves y alone and must apply to both terms on the other side.
  3. The first pair has slopes −83 and 38. Their product is −2424 = −1.This uses the slopes of the rewritten equations, rather than the coefficient 8 before y was isolated.
  4. The first pair is perpendicular.The finite nonzero slope product passes the right-angle test.
  5. The second pair is a vertical line x = 9 and a horizontal line y = −8. They meet at (9, −8).One fixes the first coordinate, and the other fixes the second coordinate.
  6. The second pair is also perpendicular. Do not form a slope product.The directions form a square corner, but the vertical line has no slope number to multiply.
Answer
  • First pair: perpendicular.
  • Second pair: perpendicular, with meeting point (9, −8).
Check For the first equation, input 1 gives y = 7, and 8 × 1 + 3 × 7 = 29 checks its rewriting. In the second pair, (9, −8) satisfies both fixed-coordinate equations and the directions are horizontal and vertical.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: y = 2x + 3 and y = −2x + 2 must be perpendicular because their slopes have opposite signs.
2 × (−2) = −4, so the turn is not a right angle.
✓ Instead: The negative reciprocal of 2 is −12, not −2.
✗ Not this: Counterexample: multiply the slopes of y = −4 and x = 7 to check perpendicularity.
The vertical slope is undefined, so a slope product cannot be formed.
✓ Instead: Their horizontal and vertical directions form a 90° angle, so they are perpendicular.
Tips and tricks
  • Tip: Equal slopes need the intercept check; opposite signs need the product check.
  • Tip: State the relationship and the numbers that prove it on an exam.
Trap. Changing only the sign or only flipping the fraction for a perpendicular slope. Both moves are needed, and the product must be −1.