Find and use equations of straight lines
Build a line equation from a gradient and point, or from two points, then interpret the result in context.
Section 7 · Straight-line graphs
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which form uses a known point and gradient directly?
What you will learn
- Find the gradient between two points
- Use point-gradient and gradient-intercept forms
- Rearrange a line equation and interpret its intercepts
Gradient measures change in y per unit change in x
For two points (x₁, y₁) and (x₂, y₂), m = (y₂ − y₁)/(x₂ − x₁), provided x₂ ≠ x₁. Keep the coordinate order consistent in both differences.
A line with gradient m through (x₁, y₁) can be written y − y₁ = m(x − x₁). The equivalent form y = mx + c shows the y-intercept c.
For a vertical line, x is constant and the gradient is undefined. It cannot be written as y = mx + c.
Choose a form that fits the information
With a point and gradient, substitute directly into point-gradient form. With two points, calculate their gradient first. Expand only if the question requests a particular form.
A point lies on a line only if its coordinates satisfy the equation. Substitution is a quick check against sign and intercept errors.
Pause and explain
What is the y-intercept of y = −3x + 7?
Worked example
Find the equation of the line through (−2, 5) with gradient 3/2.
Show the worked solution
- Use y − y₁ = m(x − x₁) with (x₁, y₁) = (−2, 5).
- y − 5 = (3/2)(x + 2).
- Expanding gives y = (3/2)x + 8.
Answer y − 5 = (3/2)(x + 2), or y = (3/2)x + 8.
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 equation through (1, 2) and (5, 10).
Give me a hint
Calculate the gradient, then use either point.
Compare my reasoning
- m = (10 − 2)/(5 − 1) = 2.
- Use y − 2 = 2(x − 1).
- Simplify to y = 2x and check both points.
y = 2x.
Look for these in your work
- Gradient calculated
- Point-gradient form used
- Both points satisfy the line
Find the line with gradient −3 through (4, −1).
Give me a hint
Substitute the point and gradient into point-gradient form.
Compare my reasoning
- Write y + 1 = −3(x − 4).
- Expand: y + 1 = −3x + 12.
- Rearrange to y = −3x + 11.
y = −3x + 11.
Look for these in your work
- Correct point substitution
- Signs preserved
- Equation simplified
A delivery service charges a fixed fee of 12 units and 7 units for each item. Write cost C as a straight-line function of item count n, then find the count when C = 61.
Give me a hint
The fixed fee is the intercept and the per-item charge is the gradient.
Compare my reasoning
- C = 7n + 12.
- Set 61 = 7n + 12, so 7n = 49.
- Therefore n = 7 items.
C = 7n + 12; a cost of 61 corresponds to 7 items.
Look for these in your work
- Fixed cost used as intercept
- Rate used as gradient
- Contextual value solved
Common mistakes
- Subtracting y-values in a different order from x-values
- Using x₁ instead of (x − x₁) in point-gradient form
- Treating a vertical line as if it had a finite gradient
What can you explain now?
What is the gradient of the line through (1, 2) and (5, 10)?
Compare with the explanation
2
The change in y is 8 and the change in x is 4, so m = 8/4.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do you find the gradient between two points and use it to write a line equation?
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.