Gradient from two stated coordinates
Use matched coordinate differences to calculate a straight line's gradient without a grid.
Extended only.
Before you begin
- Find a signed gradient from a grid.
- Subtract negative values using brackets.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is 3 − (−3)?
What you will learn
- Calculate (y₂ − y₁)/(x₂ − x₁).
- Keep the subtraction order consistent.
- Recognise horizontal or vertical lines from coordinate differences.
Coordinate differences replace the triangle
For A = (x₁, y₁) and B = (x₂, y₂), horizontal change from A to B is x₂ − x₁ and vertical change is y₂ − y₁. Dividing gives m = (y₂ − y₁)/(x₂ − x₁), provided x₂ differs from x₁. Use this when the question supplies two points rather than a grid.
For (−3, −2) and (3, 6), the changes are 3 − (−3) = 6 and 6 − (−2) = 8, so m = 8/6 = 4/3. Brackets preserve the signs of negative coordinates. The original y-values themselves are not the rise: only their difference is.
Keep the direction consistent
You may subtract A from B or B from A, but both numerator and denominator must use the same direction. Reversing both changes gives (−8)/(−6) = 4/3, the same gradient. Reversing only one gives −4/3 and incorrectly turns a rising line into a falling one.
Name or write the points in a consistent order before substituting. A negative gradient does not depend on whether the coordinates themselves are negative. For (−5, 4) and (−1, 2), y decreases by two as x increases by four, so the gradient is −1/2.
Handle a zero difference before dividing
If y₂ = y₁ but x₂ differs, the line is horizontal and m = 0. If x₂ = x₁ but y₂ differs, the line is vertical and the gradient is undefined. For (2, −1) and (2, 5), the equation is x = 2. Do not write an ordinary y = mx + c equation for it.
Two identical points do not determine one line: both differences are zero, and infinitely many lines pass through that one location. Check that the supplied points are distinct before using them to obtain a unique straight-line gradient.
Pause and explain
Which calculation uses a consistent point order?
Worked example
Find the gradient through A = (−4, 5) and B = (2, −7).
Show the worked solution
- Horizontal change B minus A is 2 − (−4) = 6.
- Vertical change is −7 − 5 = −12.
- m = −12/6 = −2; the line falls from left to right.
Answer −2
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the gradient through (−1, 2) and (3, 10).
Give me a hint
Write both differences with the second point minus the first.
Compare my reasoning
- Horizontal change is 3 − (−1) = 4.
- Vertical change is 10 − 2 = 8.
- The gradient is 8/4 = 2.
2
Look for these in your work
- Used a consistent subtraction order.
- Checked that the positive sign matches increasing y.
Find the gradient through (−5, 4) and (−1, 2).
Give me a hint
Negative x-values can still give a positive horizontal change.
Compare my reasoning
- The run is −1 − (−5) = 4.
- The rise is 2 − 4 = −2.
- m = −2/4 = −1/2.
−1/2
Look for these in your work
- Subtracted the negative coordinate correctly.
- Simplified the signed ratio.
A track passes through (4, −3) and (4, 6). Give its gradient and equation, and explain why the usual fraction cannot be evaluated.
Give me a hint
Compare the two x-values before substituting.
Compare my reasoning
- Both points have x = 4, so the horizontal change is zero.
- The vertical change is nine, but 9/0 is undefined.
- The track is the vertical line x = 4.
Undefined gradient; x = 4
Look for these in your work
- Identified the zero denominator.
- Used the fixed-coordinate equation.
Common mistakes
- Reversing only one subtraction.
- Dividing y₂ by x₂ rather than using changes.
- Treating division by zero as zero.
What can you explain now?
Find m through (1, −2) and (5, 6).
Compare with the explanation
2
The vertical change is eight and the horizontal change is four, giving 8/4 = 2.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Calculate a gradient in both point orders tomorrow and explain why both results must agree.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons; Geometry contains 15. Coordinate geometry and Geometry each have a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other five syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice and paper constructions are self-checked, not automatically graded. The checkpoints sample skills and do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources