Gradient formula: m = (y₂ − y₁) / (x₂ − x₁)
The gradient formula is gradient m = (y₂ − y₁) / (x₂ − x₁).
What each letter means
- \(m\) the gradient (slope) of the line
- \((x_1,\ y_1),\ (x_2,\ y_2)\) two points on the line
When to use it
To find how steep a line is from two points on it, before writing its equation y − y₁ = m(x − x₁).
Worked example
Find the equation of the line through \((2,5)\) that is perpendicular to the line through \((2,5)\) and \((6,7)\).
- \(m=\dfrac{7-5}{6-2}=\tfrac12\), so the perpendicular gradient is \(-2\)
- \(y-5=-2(x-2)\)
Answer: \(y=-2x+9\)
Common mistake
Putting the x difference on top. Gradient is rise over run: change in y ÷ change in x.
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) | Not in the Edexcel booklet: learn it |
| Edexcel International A Level | Not in the booklet: learn it |
| AQA or OCR A Level | Check your board’s formula booklet: AQA formulae booklet (PDF) · OCR A Level Maths assessment page |
From our own A Level Maths formula sheets, in our words. Official: Pearson's formulae booklet (PDF) · Pearson's IAL formulae booklet (PDF).
Practise and revise
- Practise A Level Maths practice
- Revise Coordinate geometry
- Revise IAL P1 coordinate geometry
- Print Edexcel A Level (9MA0) one-page formula sheet
Questions
What is the gradient formula?
The gradient formula is gradient m = (y₂ − y₁) / (x₂ − x₁). m: the gradient (slope) of the line; (x₁, y₁), (x₂, y₂): two points on the line.
Is the gradient formula given in the exam?
Edexcel A Level (9MA0): not in the Edexcel booklet: learn it. Edexcel International A Level: not in the booklet: learn it. AQA or OCR A Level: check your board’s formula booklet. This comes from our own A Level Maths formula sheets; your teacher has the official booklet.
What are the gradients of parallel and perpendicular lines?
Parallel lines have equal gradients. Perpendicular gradients multiply to −1, so a line perpendicular to gradient m has gradient −1/m.
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.