Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who must find the equation of a line from two points, a gradient and a point, or parallel/perpendicular conditions.
What query it owns: how to find and use the equation of a line in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the revision-guide angle, while Tutopiya’s Equation of a Line subtopic page owns the learning resource and the free Equation of a Line quiz owns the practice.

The equation of a line is one of the highest-frequency topics in Cambridge IGCSE Mathematics (0580/0607) Coordinate Geometry. Whether the question gives you two points, a gradient and an intercept, or a parallel/perpendicular condition, the method reduces to finding m and c in y = mx + c. This guide shows every route examiners use and the command words that signal each one.

Key takeaways

• The gradient-intercept form y = mx + c is the standard IGCSE line equation; m is gradient, c is y-intercept.
• From two points: find m first, then substitute a point to find c.
• Parallel lines share the same m; perpendicular lines have m₁ × m₂ = −1.
• Rearrange into the form requested — ax + by + d = 0 is common on mark schemes.

What is the equation of a line in Cambridge IGCSE Maths?

A straight-line equation relates x and y so every point on the line satisfies it. The form y = mx + c tells you the gradient m and where the line crosses the y-axis at (0, c). Cambridge papers ask you to find the equation from geometric information, interpret m and c in context, and find lines parallel or perpendicular to a given line.

Read the full notes on Tutopiya’s Equation of a Line subtopic page before you attempt questions.

The forms you need to know

Form	When to use	Example
y = mx + c	Gradient and y-intercept known	y = 2x − 3
y − y₁ = m(x − x₁)	Gradient and one point known	Through (3, 1) with m = 2
ax + by + d = 0	Rearranged standard form	2x − y − 7 = 0

How to find the equation of a line — step by step

1. Find the gradient m — from two points, from a given parallel line, or as the negative reciprocal for perpendicular.
2. Substitute a known point (x₁, y₁) into y = mx + c to find c.
3. Write the equation y = mx + c unless another form is requested.
4. Rearrange if the question asks for ax + by + d = 0.

Test yourself with the free Equation of a Line quiz.

Parallel and perpendicular line equations

Given	Parallel line	Perpendicular line
y = 3x + 1	Same m = 3	m = −⅓
Through (2, 5)	y − 5 = 3(x − 2)	y − 5 = −⅓(x − 2)

Equation of a line in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
Find the equation	Full line equation	”Find the equation of the line through A and B.”
Write down	State gradient or intercept	”Write down the gradient of the line 3x + 2y = 12.”
Show that	Prove a point lies on a line	”Show that (4, 5) lies on the line y = 2x − 3.”
Find the equation of the line perpendicular to	Negative reciprocal gradient	”Find the equation of the line perpendicular to y = 4x + 1 through (0, 2).”
Rearrange	Change form	”Rearrange 2y = 6x − 4 into the form y = mx + c.”

Worked exam-style stems (how to answer the wording)

1. “Find the equation of the line through (1, 3) and (5, 11).” m = (11 − 3)/(5 − 1) = 2. Using (1, 3): 3 = 2(1) + c → c = 1. Equation: y = 2x + 1.

2. “Find the equation of the line parallel to y = −2x + 5 passing through (4, 1).” Parallel → m = −2. 1 = −2(4) + c → c = 9. Equation: y = −2x + 9.

3. “Show that the line 3x + 4y = 12 crosses the y-axis at (0, 3).” Set x = 0: 4y = 12 → y = 3. Point (0, 3) satisfies the equation. Reward: substitution shown.

When you can recognise the wording instantly, work the full set on the Coordinate Geometry topical past paper questions and the Equation of a Line quiz.

How the equation of a line connects to Coordinate Geometry

Line equations depend on Distance, Midpoint and Gradient and are essential for Linear Programming boundary lines. Use the Cambridge IGCSE Maths resource hub to revise weak areas.

Common mistakes students make

• Finding c by substituting x into the wrong place in y = mx + c.
• Using the same gradient for perpendicular instead of the negative reciprocal.
• Leaving the answer as m = 2, c = 3 instead of writing y = 2x + 3.
• Misreading the gradient from ax + by = c without rearranging first.

When you need more support

If line equation questions keep costing marks, work through the Distance, Midpoint and Gradient quiz and the Coordinate Geometry topical past papers, then get help from a Cambridge IGCSE Maths tutor.

Frequently asked questions

What is y = mx + c? It is the equation of a straight line where m is the gradient and c is the y-intercept — the value of y when x = 0.

How do I find the equation from two points? Calculate m = (y₂ − y₁)/(x₂ − x₁), substitute one point into y = mx + c to find c, then write the full equation.

What is the gradient of a line perpendicular to m = 3? −⅓ — the negative reciprocal. The product of the two gradients must equal −1.

How should I revise the equation of a line? Practise all four routes (two points, gradient + point, parallel, perpendicular), then take the Equation of a Line quiz.

Ready to master Cambridge IGCSE Maths equation of a line?

Start with the Equation of a Line subtopic page, then book a free trial with a Cambridge IGCSE Maths specialist to turn line questions into guaranteed marks.

Ready to Excel in Your Studies?

Get personalised help from Tutopiya's expert tutors. Whether it's IGCSE, IB, A-Levels, or any other curriculum — we match you with the perfect tutor and your first session is free.

Book Your Free Trial