Two lines can have a special relationship: parallel means they never meet, perpendicular means they cross at a right angle. Both relationships come down to one thing — the connection between their gradients.
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What you need to know
Parallel lines have the same gradient: m1 = m2
Perpendicular lines have gradients whose product is −1: m1×m2 = −1
Perpendicular gradients are negative reciprocals — flip the fraction and change the sign
To check the relationship: rearrange both equations into y = mx + c form, then compare the gradients
Horizontal lines (y = q) and vertical lines (x = p) are perpendicular to each other
Parallel Lines
Parallel lines are equidistant — they always stay the same distance apart and never cross. The reason is simple: they have identical gradients.
If m1 = m2, then the lines are parallel
How to check if two lines are parallel
Rearrange both equations into y = mx + c form.
Read off the coefficient of x from each — that’s the gradient.
If the two gradients are equal, the lines are parallel.
Quick spot: if both equations are already in y = mx + c form, you can compare gradients straight away — no rearranging needed.
Perpendicular Lines
Perpendicular lines cross at a right angle (90°). Their gradients are negative reciprocals of each other, which means their product is exactly −1.
If m1×m2 = −1, then the lines are perpendicular
What is a “negative reciprocal”?
To find a perpendicular gradient, do two things: flip the fraction, then change the sign.
Finding the perpendicular gradient
Original
23
→
Step 1: Flip
32
→
Step 2: Sign
−32
Check: 23× (−32) = −1 ✓
Quick examples of perpendicular gradients
m = 4
→ perpendicular →
m = −14
m = 13
→ perpendicular →
m = −3
m = −52
→ perpendicular →
m = 25
Side-by-Side Comparison
Parallel Lines
m1 = m2
Same gradient — never intersect.
Perpendicular Lines
m1×m2 = −1
Cross at a right angle.
Watch out for horizontal & vertical lines: the rule m1×m2 = −1 doesn’t work here because vertical lines have an undefined gradient. But x = p and y = q ARE perpendicular — just by definition.
Worked Examples
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Example 1 — Are these lines parallel?
Are the lines y = 3x + 2 and 6x − 2y + 5 = 0 parallel?
Answer:
Step 1: rearrange the second line into y = mx + c form.6x − 2y + 5 = 02y = 6x + 5y = 3x + 5/2Step 2: compare the gradients.m₁ = 3 and m₂ = 3Yes, parallel — both have m = 3The y-intercepts differ, so they are not the same line.
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Example 2 — Equation of a parallel line
The line l passes through the point (4, −1) and is parallel to the line with equation 2x − 5y = 3. Find the equation of l in the form y = mx + c.
Answer:
Step 1: rearrange the given line to find its gradient.2x − 5y = 35y = 2x − 3y = (2/5)x − 3/5gradient = 2/5Step 2: parallel ⇒ same gradient. So m = 2/5.Step 3: use point-gradient form with (4, −1).y − (−1) = (2/5)(x − 4)y + 1 = (2/5)x − 8/5y = (2/5)x − 8/5 − 1y = (2/5)x − 8/5 − 5/5y = (2/5)x − 13/5
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Example 3 — Find the perpendicular gradient
A line has gradient −34. Find the gradient of any line perpendicular to it.
The line l1 has equation 3x − 5y = 7. The line l2 has equation y = 14 − 53x. Determine whether l1 and l2 are perpendicular.
Answer:
Step 1: rearrange l₁ into y = mx + c.3x − 5y = 75y = 3x − 7y = (3/5)x − 7/5m₁ = 3/5Step 2: read m₂ from l₂ (already in y = mx + c).m₂ = −5/3Step 3: multiply the gradients.m₁ × m₂ = (3/5) × (−5/3) = −15/15 = −1Yes, perpendicular — m₁ × m₂ = −1
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Example 5 — Equation of a perpendicular line
Find the equation of the line that is perpendicular to y = 2x + 1 and passes through the point (4, 3). Give your answer in the form y = mx + c.
Answer:
Step 1: read the original gradient.m₁ = 2Step 2: find the perpendicular gradient (negative reciprocal).flip: 1/2 → change sign: −1/2m₂ = −1/2Step 3: use point-gradient form with (4, 3).y − 3 = −(1/2)(x − 4)y − 3 = −(1/2)x + 2y = −(1/2)x + 2 + 3y = −(1/2)x + 5Check: 2 × (−1/2) = −1 ✓ — perpendicular confirmed.
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Tips
Always rearrange first. If the equation isn’t in y = mx + c form, rearrange before doing anything else.
Quick check trick: if you suspect two lines are perpendicular, just multiply the gradients. If you get exactly −1, you’re right.
For perpendicular gradients, remember the two-step rule: flip and change the sign. Don’t just flip OR change the sign — you must do both.
Whole numbers count too: a gradient of 4 is really 41. Flip to 14, change sign → −14.
State your reason: when asked to “determine and explain”, always finish with “since m1 = m2” or “since m1×m2 = −1″ — examiners look for this.
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Common mistakes
Forgetting to change the sign when finding a perpendicular gradient. 23 flipped is 32 — but the perpendicular gradient is −32, not 32.
Confusing parallel and perpendicular rules. Parallel = same gradient. Perpendicular = product is −1. Mixing them up is the most common slip.
Comparing gradients before rearranging. You can’t read the gradient from 2x − 5y = 3 directly — you must isolate y first.
Reciprocal ≠ negative reciprocal. The reciprocal of 2 is 12; the perpendicular gradient is −12. Don’t drop the minus.
Sign errors in fractions: watch out when the original gradient is already negative — the perpendicular gradient ends up positive.
Stopping at the gradient. If the question asks for the equation, you still need to use point-gradient form and rearrange to the requested form.
Final word: Two simple rules — m1 = m2 for parallel, m1×m2 = −1 for perpendicular — unlock every parallel/perpendicular question in IB. Get these in your head and you’re sorted.
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