Solving Linear Equations

What a linear equation is

A linear equation contains a letter (the unknown) to the power 1 — no squares or higher powers. Solving it means finding the value of the unknown that makes both sides equal. The golden rule: whatever you do to one side, you must do to the other to keep it balanced.

The balance method

Work step by step to get the unknown on its own (isolate it), using inverse operations:

• The inverse of + is − (and vice versa).
• The inverse of × is ÷ (and vice versa).

Example: solve 3x + 5 = 20.

1. Subtract 5 from both sides: 3x = 15.

2. Divide both sides by 3: x = 5.

Check: 3(5) + 5 = 20 ✓

Unknowns on both sides

Collect the unknowns on one side and the numbers on the other.

Example: solve 5x − 2 = 3x + 8.

1. Subtract 3x from both sides: 2x − 2 = 8.

2. Add 2 to both sides: 2x = 10.

3. Divide by 2: x = 5. ✓

Equations with brackets

Expand the brackets first, then solve.

Example: solve 2(x + 4) = 18.

1. Expand: 2x + 8 = 18.

2. Subtract 8: 2x = 10.

3. Divide by 2: x = 5. ✓

Equations with fractions

Multiply every term by the denominator to remove the fraction.

• x/4 = 3 → multiply by 4 → x = 12.

Forming equations from words

Turn a written problem into an equation, then solve. E.g. "I think of a number, multiply by 3 and add 4 to get 19": 3n + 4 = 19 → n = 5.

Worked example

Solve 4(2x − 1) = 28.

1. Expand: 8x − 4 = 28.

2. Add 4: 8x = 32.

3. Divide by 8: x = 4. Check: 4(2×4 − 1) = 4×7 = 28 ✓

Common mistakes

• Only doing an operation to one side — always do it to both.
• Forgetting to expand brackets before collecting terms.
• Sign errors when moving negative terms across.

Exam tips

• Show every step — you get method marks even if the final answer slips.
• Always check your answer by substituting it back in.
• For worded questions, define your letter first (e.g. "let the number be n").

Key facts to remember

• Keep the equation balanced: do the same to both sides, using inverse operations.
• Expand brackets and clear fractions first, then collect unknowns on one side.
• Always substitute back to check.