Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Straight Lines: y = mx + c

In one sentence. Gradient as a rate of climb, intercept as the starting height — and building a line's equation from any two of its points.

Coordinate Geometry — one of 2 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with René Descartes. Learn Straight Lines: y = mx + c, teach it, then defend it against René Descartes.

Where this sits in the specification

Specification 4MA1, 3.3 Graphs — linear. Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 3.3 — Graphs — linear

1 · Understand it

What it is

Every non-vertical straight line is y = mx + c: m is the gradient (how much y climbs when x increases by 1) and c is the y-intercept (the height where the line crosses the y-axis). Two points determine the line completely: gradient from the rise over the run, then c from either point.

Why it works

A line is exactly the shape with a CONSTANT rate of climb — that's what makes it straight. Gradient = rise ÷ run because 'per unit of x' means dividing the vertical change by the horizontal change. Once m is known, c is forced: the line must pass through your known point, so substitute it and solve.

When a question is asking for it

Reading real graphs (m is a speed, a price per unit, a rate); 'find the equation of the line through…' questions; and as the foundation for parallel/perpendicular work and linear modelling.

How to hold it in your head

m tells you the story per step: m = 3 means 'every step right, three steps up'. Negative m walks downhill. c is where the story starts — on the y-axis. y = 3x + 1 in words: 'start at 1, climb 3 per step'.

2 · The mistakes that cost marks

  • Gradient as run over rise. It's rise OVER run — vertical change divided by horizontal change.
  • Reading m and c from an equation not yet in y = form: 2y = 6x + 4 has gradient 3, not 6. Rearrange first.
  • Subtracting coordinates in different orders top and bottom: (y₂ − y₁)/(x₂ − x₁) must keep the points in the SAME order.
  • Assuming c can be read from any point. c is the y-value only where x = 0.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

Find the equation of the line through (2, 7) and (5, 16).

  1. Gradient: m = (16 − 7)/(5 − 2) = 9/3 = 3.
  2. So y = 3x + c. Substitute (2, 7): 7 = 6 + c → c = 1.
  3. Equation: y = 3x + 1. (Check with the other point: 3(5) + 1 = 16 ✓)

Answer: y = 3x + 1

Find the gradient of the line through (1, 3) and (4, 15).

  1. Rise: 15 − 3 = 12. Run: 4 − 1 = 3.
  2. m = 12/3 = 4.

Answer: 4

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — Work out the gradient of the line passing through (1, 3) and (4, 15).
  2. Grade 7 — Write down the equation of the line with gradient 2 passing through (0, −3). (Type it like y=2x-3.)
  3. Grade 8 — Find the equation of the line through (2, 7) and (5, 16). (Type it like y=3x+1.)
  4. Grade 9 — A = (−2, 5) and B = (6, −1). Work out the midpoint of AB. (Type it like (2,2).)
  5. 9+ — A = (1, 2) and B = (7, 10). Work out the length of AB.

Ask the person who made it

Inside Incandio you can put a question to Katherine Johnson, who answers knowing exactly which module you have just finished: “You computed spacecraft trajectories by hand — gradients, intercepts, coordinates, all of it. How did the coordinate geometry we learn at school become the mathematics that brought John Glenn home safely?”

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Straight Lines: y = mx + c, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Straight Lines: y = mx + c works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Straight Lines: y = mx + c against René Descartes — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 8 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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