Build a straight-line equation from points and gradients
Choose a useful line form, retain signs and recognise when a vertical line cannot use a finite gradient.
Pure Mathematics 1 (Paper 1) · AS & A Level Mathematics 9709, 2026–2027. Use algebra and elementary geometry; implicit differentiation is outside this section. Written practice is self-checked.
Before you begin
- Subtract signed coordinates and solve linear equations.
- Interpret a coordinate pair as an input and output.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is 4 − (−2)?
What you will learn
- Find a line from two distinct points or one point and a gradient.
- Convert between gradient-intercept, point-gradient and general line forms.
Use the same point order in both differences
For distinct points (x₁, y₁) and (x₂, y₂) with x₂ ≠ x₁, the gradient is m = (y₂ − y₁)/(x₂ − x₁). Reversing both differences gives the same quotient; reversing just one changes its sign. A known point and gradient give y − y₁ = m(x − x₁). Substitute the point to check the equation.
For A(−2, 5) and B(4, −1), m = (−1 − 5)/(4 − (−2)) = −1. Using A gives y − 5 = −(x + 2), hence y = −x + 3. Both points satisfy it. Distinct points with equal x-coordinates define a vertical line x = constant; its gradient is undefined, rather than zero. Equal y-coordinates give a horizontal line with gradient zero.
Choose a form that shows what you need
In y = mx + c, m is the gradient and c the y-intercept. In y − y₁ = m(x − x₁), the given point and gradient remain visible. In ax + by + c = 0, at least one of a and b must be non-zero. If b ≠ 0, rearranging gives gradient −a/b and y-intercept −c/b. If b = 0, the equation is vertical and must be treated separately.
Multiplying a line equation by any non-zero constant does not change its points, so x + y − 3 = 0 and 2x + 2y − 6 = 0 describe the same line. A data-based linear model assumes a constant rate over its stated domain. Finding a line through two observations does not by itself prove that a real process is linear beyond them.
Pause and explain
What is the gradient of 3x + 2y − 8 = 0?
Worked example
Find the line through (−2, 5) and (4, −1), giving gradient-intercept and general forms.
Show the worked solution
- The gradient is (−1 − 5)/(4 + 2) = −6/6 = −1.
- Use y − 5 = −(x + 2), then simplify to y = −x + 3.
- Move all terms to one side: x + y − 3 = 0; substitute both original points to check.
Answer y = −x + 3, or x + y − 3 = 0.
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 line through (3, −2) with gradient 2.
Give me a hint
Use y − y₁ = m(x − x₁) before expanding.
Compare my reasoning
- Substitute the point and gradient: y + 2 = 2(x − 3).
- Expand and rearrange to y = 2x − 8.
- At x = 3, the equation gives y = −2, confirming the given point.
y = 2x − 8.
Look for these in your work
- Use the signed point correctly.
- Check the line contains the given point.
Find the line through (−1, −3) and (5, 9).
Give me a hint
Calculate a signed gradient and then use either point.
Compare my reasoning
- The gradient is (9 + 3)/(5 + 1) = 12/6 = 2.
- Using (−1, −3), write y + 3 = 2(x + 1), giving y = 2x − 1.
- The other point gives 2(5) − 1 = 9, so both points lie on the line.
y = 2x − 1.
Look for these in your work
- Use consistent coordinate differences.
- Verify the second point after finding the equation.
A supplied linear model passes through (2, 7) and (6, 19) for 2 ≤ x ≤ 6. Find its rule and the modelled value at x = 4.
Give me a hint
Find the constant rate, then determine the intercept from an observation.
Compare my reasoning
- The rate is (19 − 7)/(6 − 2) = 3.
- Using 7 = 3(2) + c gives c = 1, so y = 3x + 1 on the stated interval.
- The permitted input 4 gives y = 13; the stated model does not justify predictions outside its domain.
y = 3x + 1 for 2 ≤ x ≤ 6; y(4) = 13.
Look for these in your work
- Find the rule from both supplied points.
- Respect the model's domain when interpreting the result.
Common mistakes
- Keep the same order in the y- and x-differences.
- An undefined vertical gradient is different from a zero horizontal gradient.
- Read −a/b as the gradient of ax + by + c = 0 only when b ≠ 0.
What can you explain now?
Find the line through (4, −2) and (4, 9).
Compare with the explanation
x = 4; its gradient is undefined.
Both points have x-coordinate 4. The zero denominator in the gradient calculation indicates a vertical line.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Give three equivalent forms of one non-vertical line, then explain how a vertical line is represented and why its gradient is undefined.
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.