Linear Equations

What is a Linear Equation?

An equation where the highest power of the variable is 1 — its graph is always a straight line.

ax + b = 0 a, b are constants · a ≠ 0

No x², √x or 1 x terms — just x to the power 1.

One variable → exactly one solution.

Solving by Balancing

Keep the equation balanced: whatever you do to one side, do to the other. Undo operations in reverse.

1 Move number terms away from x: 2x + 5 = 11 → 2x = 6 (−5 both sides)

2 Undo the multiply: x = 6 ÷ 2 = 3

3 Check: 2(3) + 5 = 11 ✓

Brackets & x on Both Sides

Expand brackets first, then collect the x-terms on one side.

3(x + 2) = 2x + 11

1 Expand: 3x + 6 = 2x + 11

2 x to one side: 3x − 2x = 11 − 6

3 Simplify: x = 5

The Line y = mx + c

m = gradient (steepness) · c = y-intercept (where it crosses the y-axis).

y = 2x + 1 → gradient 2, crosses y-axis at 1

m > 0 → uphill · m < 0 → downhill Bigger |m| → steeper line.

Finding the Gradient

From two points (x₁, y₁) and (x₂, y₂):

m = rise run = y₂ − y₁ x₂ − x₁

e.g. through (1, 3) and (4, 9):

m = 9 − 3 4 − 1 = 6 3 = 2

Worked Example

Find the equation of the line through (0, −1) with gradient 3, then solve where it meets y = 5.

1 Gradient m = 3, y-intercept c = −1 → y = 3x − 1

2 Set equal to 5: 3x − 1 = 5

3 Add 1: 3x = 6

4 Divide by 3: x = 2 → meets at (2, 5)

Exam tip Always substitute your answer back in: 3(2) − 1 = 5 ✓. One mark, every time.

Owlileo · IGCSE Mathematics (0580) · Algebra 2.5 — Linear Equations