Linear Equations

Algebra • Topic 7 • intermediate

A linear equation is like a mathematical balance scale. Both sides must be equal, and we can solve for the unknown variable.

ax + b = c

Where a, b, and c are numbers, and x is the variable we need to find

x + 3 = 7

Simple equation

2x - 5 = 9

With coefficient

3x + 4 = 2x + 7

Variables on both sides

The key rule: whatever you do to one side, you must do to the other side to keep it balanced!

x + 5 = 12 (original equation)

x + 5 - 5 = 12 - 5 (subtract 5 from both sides)

x = 7 (solution)

3x - 4 = 14 (original equation)

3x - 4 + 4 = 14 + 4 (add 4 to both sides)

3x = 18 (simplify)

x = 6 (divide both sides by 3)

When variables appear on both sides, collect them on one side first.

5x + 3 = 2x + 12 (original)

5x - 2x + 3 = 2x - 2x + 12 (subtract 2x from both sides)

3x + 3 = 12 (simplify)

3x = 9 (subtract 3 from both sides)

x = 3 (divide by 3)

Always substitute your answer back into the original equation to check!

Left side: 5(3) + 3 = 15 + 3 = 18

Right side: 2(3) + 12 = 6 + 12 = 18

✓ Both sides equal 18, so x = 3 is correct!

1. Solve: x + 8 = 15

2. Solve: 2x - 3 = 11

3. Solve: 4x + 1 = 2x + 9