Gradient from two stated coordinates
Use matched coordinate differences to calculate a straight line's gradient without a grid.
Section Paper 2 · Extended · Non-calculator
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
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
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →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.