How do you find the gradient of a straight line from two points?
Short answer
The gradient is the change in y divided by the change in x. For points (x₁, y₁) and (x₂, y₂), gradient m = (y₂ − y₁) / (x₂ − x₁). A positive gradient slopes upwards from left to right, a negative gradient slopes downwards.
Step by step
- Label the two points (x₁, y₁) and (x₂, y₂). It does not matter which is first, as long as you stay consistent.
- Subtract the y values: y₂ − y₁.
- Subtract the x values in the same order: x₂ − x₁.
- Divide the change in y by the change in x and simplify the fraction.
Worked example
Find the gradient of the line through (1, 3) and (4, 12)
- Change in y: 12 − 3 = 9
- Change in x: 4 − 1 = 3
- Gradient = 9 / 3 = 3
Common mistakes
- Putting the change in x on top. Remember: rise over run, y over x.
- Subtracting in a different order on the top and bottom, which flips the sign.
- Losing a minus sign when one of the coordinates is negative. Use brackets: 5 − (−2) = 7.
Useful for:GCSEBECEBEPCCheckpoint
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