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Substitution

FoundationHigherAQAEdexcelOCR

Practise substitution with this free Foundation and Higher GCSE Maths worksheet from Virtus Academy. You'll work through substituting into expressions and formulae, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Replace each letter with its value and apply BIDMAS, watching the signs.

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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

Substitution means replacing letters in a formula or expression with given numbers, then evaluating the result.

The critical detail is order of operations. In \(3x^2\) with \(x = 4\), you square first and then multiply, giving \(48\), not \(144\). The index applies only to the \(x\), not to the \(3x\).

Negative values need extra care. Substituting \(x = -3\) into \(x^2\) gives \(9\), because a negative squared is positive, but into \(-x^2\) gives \(-9\). Writing the substituted value in brackets makes the intention unambiguous and prevents most errors.

Revision notes

Substituting carefully

Replace each letter with its value, using brackets around negatives, then apply the order of operations.

For \(2a + 3b\) with \(a = 5\) and \(b = -2\): \(2(5) + 3(-2) = 10 - 6 = 4\).

Powers and coefficients

An index attaches only to the letter immediately before it, unless brackets say otherwise.

So \(3x^2\) with \(x = 4\) is \(3 \times 16 = 48\). By contrast \((3x)^2\) would be \(12^2 = 144\).

Substituting into formulae

Formulae from other topics work the same way. Write the formula, substitute, then evaluate.

For the area of a triangle with \(b = 8\) and \(h = 5\): \(A = \frac{1}{2}bh = \frac{1}{2}(8)(5) = 20\). Include units in the final answer where the context requires them.

Key points

  • Replace each letter with its given value.
  • Use brackets around negative values.
  • Apply the order of operations after substituting.
  • An index applies only to the letter it follows.
  • A negative squared gives a positive.
  • Include units when substituting into a formula.

Worked examples

Example 1

Find the value of \(4x + 7\) when \(x = -3\).

Working

\[4(-3) + 7\]substitute using brackets
\[-12 + 7\]multiply first
\[= -5\]then add

Example 2

Find the value of \(2x^2\) when \(x = 5\).

Working

\[2 \times 5^2\]the index applies only to the x
\[2 \times 25\]square before multiplying
\[= 50\]evaluate the result

Example 3

Find the value of \(3a - b^2\) when \(a = 4\) and \(b = -2\).

Working

\[3(4) - (-2)^2\]substitute both values using brackets
\[12 - 4\]a negative squared is positive
\[= 8\]subtract to give the answer

Common mistakes

  • Multiplying before applying the index.

    3x² with x = 4 is 48, not 144. Square the x first.

  • Making a squared negative stay negative.

    (−2)² is 4, not −4. Brackets make this clear.

  • Dropping the brackets around a negative.

    Writing 4 × −3 without brackets often leads to sign errors in longer expressions.

  • Ignoring the order of operations.

    Substitution does not change BIDMAS; it still applies to the numbers.

Exam tips

  • Write brackets around every substituted value, especially negatives.
  • Apply indices before multiplication.
  • Set the calculation out over several lines.
  • Include units when the formula describes a real quantity.

Key terms

Substitute
To replace a letter with a number.
Formula
A rule connecting quantities, written in symbols.
Evaluate
To work out the numerical value.
Index
A power showing repeated multiplication.

Related topics

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