The gradient of a line measures how steeply it rises or falls: change in y divided by change in x. It appears whenever a question gives two points and asks about the line joining them, and it feeds every other lesson in coordinate geometry.
What does the gradient formula say?
For two points (x₁, y₁) and (x₂, y₂), the gradient is m = (y₂ − y₁) / (x₂ − x₁). The numerator is how far you go up or down. The denominator is how far you go across.
A positive gradient means the line rises as you move right. A negative gradient means it falls. A gradient of 0 is a flat line, and a line with no gradient value at all is vertical.
How do you work it out, step by step?
- Label the points. Call one (x₁, y₁) and the other (x₂, y₂). It does not matter which is which.
- Subtract the y-values in that order to get the rise.
- Subtract the x-values in the same order to get the run.
- Divide rise by run and simplify the fraction.
- Sense-check the sign by sketching both points: does the line go up or down as you move right?
Worked example
Find the gradient of the line through P(−3, 4) and Q(5, −8).
Step 1, label: (x₁, y₁) = (−3, 4) and (x₂, y₂) = (5, −8).
Step 2, rise: y₂ − y₁ = −8 − 4 = −12.
Step 3, run: x₂ − x₁ = 5 − (−3) = 5 + 3 = 8.
Step 4, divide: m = −12/8 = −3/2.
Step 5, sense-check: P is on the left and high up, Q is on the right and low down. The line falls as you move right, so a negative gradient is correct.
Independent check: swap the labels. Then the rise is 4 − (−8) = 12 and the run is −3 − 5 = −8, giving 12/(−8) = −3/2. Same answer.
The mistake to watch for
The usual slip is handling the minus signs badly in the run.
Mistaken working: run = 5 − 3 = 2, so m = −12/2 = −6.
The student dropped the negative sign of the x-value −3 and wrote 3 instead.
Subtracting a negative means adding: 5 − (−3) = 8, not 2. Put brackets round every negative coordinate before you subtract, and the slip disappears. The sketch also helps, because a gradient of −6 would be far steeper than the picture shows.
A second slip is mixing the order, for example −8 − 4 on top but 3 − 5 on the bottom. Keep both subtractions in the same direction.
Check yourself
Try these without a calculator, then open each answer.
1. Find the gradient through (1, 2) and (4, 11).
Show answer
Rise = 11 − 2 = 9. Run = 4 − 1 = 3. Gradient = 9/3 = 3.
2. Find the gradient through (−2, 5) and (6, 1).
Show answer
Rise = 1 − 5 = −4. Run = 6 − (−2) = 8. Gradient = −4/8 = −1/2.
3. What happens when you try to find the gradient through (3, −1) and (3, 7)?
Show answer
Run = 3 − 3 = 0, so you would divide by 0. The gradient is undefined: the line is vertical, with equation x = 3.
Where this leads next
Once gradients feel secure, move on to finding a midpoint and a segment length, then use a gradient to write the equation of a line. The non-calculator working trainer is handy for practising fraction simplification without a calculator, and the coordinate geometry practice set mixes all the skills together.
Some students know the formula but still lose marks on signs once the coordinates turn negative. In online one-to-one Mathematics tuition, a teacher can watch your subtraction line by line and fix the habit that flips the sign.