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Solving Equations

FoundationHigherAQAEdexcelOCR

Solving Equations is a key algebra topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on solving linear equations, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Keep the equation balanced — whatever you do to one side, do to the other.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Solving an equation means finding the value of the unknown that makes both sides equal. You do this by undoing the operations applied to the unknown, working in reverse order.

Whatever you do to one side you must do to the other. That is what keeps the equation balanced, and it is the single rule underpinning every step.

When the unknown appears on both sides, collect it on one side first. Move the smaller quantity of the unknown to avoid negatives, so in \(5x + 2 = 3x + 10\) subtract \(3x\) from both sides rather than \(5x\).

Revision notes

Undoing operations

Work backwards through the operations, doing the opposite each time.

For \(3x + 5 = 20\): subtract \(5\) from both sides to get \(3x = 15\), then divide both sides by \(3\) to get \(x = 5\).

Unknowns on both sides

Collect the unknown on one side by adding or subtracting the same term from both sides.

For \(7x - 3 = 4x + 9\): subtract \(4x\) to get \(3x - 3 = 9\), then add \(3\) and divide by \(3\), giving \(x = 4\).

Equations with brackets or fractions

Expand any brackets first, or multiply through to clear fractions, then solve as usual.

For \(\frac{x}{4} + 2 = 7\): subtract \(2\) to get \(\frac{x}{4} = 5\), then multiply both sides by \(4\), giving \(x = 20\).

Key points

  • Do the same to both sides to keep the equation balanced.
  • Undo operations in reverse order.
  • Collect the unknown on one side when it appears on both.
  • Expand brackets before solving.
  • Multiply through to clear fractions.
  • Check by substituting your answer back.

Worked examples

Example 1

Solve \(4x - 7 = 21\).

Working

\[4x = 28\]add 7 to both sides
\[x = 7\]divide both sides by 4

Example 2

Solve \(6x + 5 = 2x + 17\).

Working

\[4x + 5 = 17\]subtract 2x from both sides
\[4x = 12\]subtract 5 from both sides
\[x = 3\]divide both sides by 4

Example 3

Solve \(3(x + 2) = 18\).

Working

\[3x + 6 = 18\]expand the bracket
\[3x = 12\]subtract 6 from both sides
\[x = 4\]divide both sides by 3

Common mistakes

  • Doing something to one side only.

    The equation must stay balanced, so every operation applies to both sides.

  • Undoing operations in the wrong order.

    In 3x + 5 = 20, subtract the 5 before dividing by 3.

  • Sign errors when moving terms.

    Subtracting 4x from both sides of 7x − 3 = 4x + 9 leaves 3x − 3 = 9.

  • Not checking the answer.

    Substituting back takes seconds and confirms both sides are equal.

Exam tips

  • Write each step on a new line with the operation shown at the side.
  • Collect the unknown on the side that keeps it positive.
  • Expand brackets and clear fractions before anything else.
  • Substitute your answer back into the original equation to check.

Key terms

Equation
A statement that two expressions are equal.
Solve
To find the value of the unknown.
Inverse operation
The operation that undoes another.
Balance
Keeping both sides equal by doing the same to each.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.