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 understand the formula but still lose marks on signs once the coordinates turn negative. That is the kind of pattern our teachers look for in online one-to-one Mathematics tuition.