Linear Equations

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To solve a linear equation, undo the operations on the unknown using inverse operations, doing the same to both sides each time. For equations with brackets, expand first or divide by the number outside. When the unknown is on both sides, collect the unknown terms on one side and the numbers on the other, then solve as usual.

Solving strategies

Do the same operation to both sides
Two-step: undo + or − first, then × or ÷
Brackets: expand, or divide by the number outside
Unknown on both sides: collect unknowns on the side with more of them
Fractions: multiply both sides by the denominator

Solving by inverse operations

An equation is balanced, so whatever you do to one side you must do to the other. For 3x + 7 = 25, subtract 7 from both sides to get 3x = 18, then divide by 3 to get x = 6. A function machine shows the operations applied to x; solving runs them backwards.

Equations with brackets

Either expand the bracket first or divide both sides by the number outside. For 4(x − 2) = 24, dividing by 4 gives x − 2 = 6, so x = 8. Expanding gives 4x − 8 = 24, so 4x = 32 and x = 8 again. Dividing first is quicker when it gives whole numbers.

Unknowns on both sides

Collect the unknown terms on one side, usually the side with more of them, so the coefficient stays positive. For 6x − 4 = 2x + 32, subtract 2x from both sides to get 4x − 4 = 32, then add 4 and divide by 4: x = 9.

Fractions and negative answers

If the unknown is part of a fraction, multiply both sides by the denominator first. For (x + 3)/4 = 5, multiply by 4 to get x + 3 = 20, so x = 17. Answers can be negative or fractional, so do not assume something has gone wrong if x is not a positive whole number.

Forming and solving equations

Many questions describe a situation, such as a perimeter or an angle sum, and ask you to form an equation. Write an expression for the quantity, set it equal to the given value, then solve. Check that your answer makes sense in context.

Worked examples

Example 1. Solve 3x + 7 = 25.

Subtract 7: 3x = 18.
Divide by 3.

Answer: x = 6.

Example 2. Solve 4(x − 2) = 24.

Divide by 4: x − 2 = 6.
Add 2.

Answer: x = 8.

Example 3. Solve 6x − 4 = 2x + 32.

Subtract 2x: 4x − 4 = 32.
Add 4: 4x = 36.
Divide by 4.

Answer: x = 9.

Example 4. Solve (x + 3)/4 = 5.

Multiply by 4: x + 3 = 20.
Subtract 3.

Answer: x = 17.

Example 5. Solve 3x + 14 = 5.

Subtract 14: 3x = −9.
Divide by 3.

Answer: x = −3.

Example 6. A rectangle has length 2x and width x + 3. Its perimeter is 42 cm. Find x.

Perimeter = 2(2x) + 2(x + 3) = 6x + 6.
6x + 6 = 42, so 6x = 36.

Answer: x = 6.

Common mistakes

Doing an operation to one side only. Keep the equation balanced by applying every step to both sides.
Undoing in the wrong order. For 3x + 7 = 25, subtract 7 before dividing by 3.
Sign errors when moving terms. Subtracting 2x from both sides removes it from the right and reduces the left.
Multiplying only part of a fraction equation. Multiply every term on both sides by the denominator.
Not checking the answer. Substitute your solution back into the original equation.

How linear equations are examined

Solving linear equations is assessed on every tier of the Edexcel GCSE (1MA1) and the 4MA1 International GCSE, both as stand-alone questions and inside geometry and problem-solving questions. You may need to: