Linear Equations

Linear equations are the foundation of IGCSE 0580 algebra — everything else builds on them. The highest power of the variable is 1, and the graph is always a straight line.

Solving an equation

The goal is to isolate the variable (usually x) by doing the same operation to both sides:

3x + 5 = 20

3x = 20 − 5 = 15 (subtract 5 from both sides)

x = 15 ÷ 3 = 5 (divide both sides by 3)

If there are brackets, expand first: 2(x − 4) = 10 → 2x − 8 = 10 → x = 9.

The equation of a line: y = mx + c

• m = gradient: the steepness of the line. m = (y₂ − y₁) / (x₂ − x₁)
• c = y-intercept: where the line crosses the y-axis.

Example: a line through (0, 2) and (4, 10) →

gradient m = (10 − 2)/(4 − 0) = 2, intercept c = 2 → y = 2x + 2.

Worked example

A line passes through (1, 5) and (3, 11). Find its equation.

1. Gradient: m = (11 − 5)/(3 − 1) = 6/2 = 3
2. Substitute one point: 5 = 3(1) + c → c = 2
3. Answer: y = 3x + 2

Exam tip

In 0580 there are marks for "Show your working": write every step. It is not only the final answer that scores — setting up the gradient and intercept correctly earns marks too. For graph questions, finding two points and drawing with a ruler is the safest method.