Two lines are parallel when they have the same gradient. They are perpendicular when their gradients multiply to −1. Exam questions use this to ask whether lines are parallel, or to write the equation of a line parallel or perpendicular to a given one through a given point.
What are the two rules?
- Parallel: m₁ = m₂.
- Perpendicular: m₁ × m₂ = −1. Equivalently, the second gradient is the negative reciprocal of the first: flip the fraction, then change the sign.
The negative reciprocal of 3/2 is −2/3. The negative reciprocal of −5 is 1/5. Always check by multiplying: (3/2) × (−2/3) = −1, and (−5) × (1/5) = −1.
How do you work it out, step by step?
- Rearrange to y = mx + c so the gradient is visible.
- Read off m. Parallel lines keep it; perpendicular lines use the negative reciprocal.
- Use the given point with y − y₁ = m(x − x₁), as in writing the equation of a line.
- Check by substituting the point, and by confirming the gradient rule.
Worked example
The line L has equation 2x + 3y = 6. Find (a) the equation of the line parallel to L through (3, 1), and (b) the equation of the line perpendicular to L through (4, −1).
Step 1, gradient of L: 3y = −2x + 6, so y = −2/3 x + 2. The gradient is m = −2/3.
(a) Parallel: the gradient is also −2/3. Then y − 1 = −2/3(x − 3), so y − 1 = −2/3 x + 2, giving y = −2/3 x + 3. Check: at x = 3, y = −2 + 3 = 1. ✓
(b) Perpendicular: the gradient is the negative reciprocal of −2/3, which is 3/2. Then y − (−1) = 3/2(x − 4), so y + 1 = 3/2 x − 6, giving y = 3/2 x − 7. Check: at x = 4, y = 6 − 7 = −1. ✓
Gradient check: (−2/3) × (3/2) = −1, so the lines are perpendicular. ✓
The mistake to watch for
The usual slip is doing only half of the rule.
Mistaken working: perpendicular gradient of −2/3 is −3/2.
The student flipped the fraction but did not change the sign. The product is (−2/3) × (−3/2) = +1, not −1.
The product test catches it at once; flip and change the sign, so −2/3 becomes 3/2. A second slip is reading the gradient from 2x + 3y = 6 as 2 or −2, without rearranging first. The gradient is the coefficient of x only once y stands alone.
Check yourself
Try these without a calculator, then open each answer.
1. Are y = 4x + 1 and x + 4y = 8 perpendicular?
Show answer
The second line: 4y = −x + 8, so y = −1/4 x + 2, with gradient −1/4. The product 4 × (−1/4) = −1, so yes, they are perpendicular.
2. What is the gradient of any line perpendicular to a line with gradient 5?
Show answer
Flip 5 (which is 5/1) to get 1/5, then change the sign: −1/5. Check: 5 × (−1/5) = −1.
3. Is 3x − y = 7 parallel to y = 3x + 2?
Show answer
Rearrange: y = 3x − 7. The gradient is 3, the same as the other line, and the intercepts (−7 and 2) differ. So yes, parallel and distinct.
Where this leads next
Try the last skill in this module, checking whether three points are collinear, which uses equal gradients in a new way. Return to gradient from two points if you need a refresher, then test yourself with the coordinate geometry practice set.
If you have the rule memorised but mix up the sign under exam pressure, that detail is exactly what a teacher can fix in online one-to-one Mathematics tuition.