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Identities

FoundationHigherAQAEdexcelOCR

This free Foundation and Higher GCSE Maths worksheet on identities helps you revise proving and using identities. Questions build from straightforward to exam standard, with full worked answers in the mark scheme — ideal for revision or homework. Suitable for AQA and OCR. An identity is true for every value, shown with the ≡ sign.

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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

An identity is an equation that is true for every value of the unknown, not just for particular values. It is written with the identity symbol, three horizontal lines, rather than an ordinary equals sign.

So \(2(x + 3) \equiv 2x + 6\) is an identity, because expanding the left side gives exactly the right side whatever \(x\) is. By contrast \(2x + 6 = 10\) is an equation, true only when \(x = 2\).

Questions usually give an identity with unknown coefficients and ask you to find them. The technique is to expand one side and compare coefficients of matching terms, since the two sides must agree term by term.

Revision notes

Identity versus equation

An equation is true for particular values; an identity is true for all values.

The symbol \(\equiv\) signals an identity. Recognising which you are dealing with tells you whether to solve or to compare coefficients.

Comparing coefficients

Expand one side fully, then match the coefficients of each power of \(x\) and the constants.

If \(a(x + 2) \equiv 3x + b\), expanding gives \(ax + 2a\). Comparing the \(x\) terms gives \(a = 3\), and comparing constants gives \(b = 2a = 6\).

Proving an identity

Take one side and manipulate it until it matches the other. Never work on both sides at once.

Starting from the more complicated side is usually easier, and every step must be reversible and clearly shown.

Key points

  • An identity is true for all values of the unknown.
  • The symbol is \(\equiv\), with three lines.
  • An equation is true only for particular values.
  • Expand one side and compare coefficients.
  • Match the \(x\) terms and the constants separately.
  • Work on one side only when proving an identity.

Worked examples

Example 1

Find \(a\) and \(b\) if \(a(x + 4) \equiv 5x + b\).

Working

\[ax + 4a \equiv 5x + b\]expand the left-hand side
\[a = 5\]compare the coefficients of x
\[b = 4a = 20\]compare the constants

Example 2

Find \(p\) if \(3(2x - p) \equiv 6x - 15\).

Working

\[6x - 3p \equiv 6x - 15\]expand the left-hand side
\[-3p = -15\]compare the constant terms
\[p = 5\]solve for p

Example 3

Explain the difference between \(2x + 4 = 10\) and \(2(x + 2) \equiv 2x + 4\).

Working

\[\text{The first is true only when } x = 3\]an equation holds for particular values
\[\text{The second is true for every } x\]an identity holds for all values

Common mistakes

  • Treating an identity as an equation to solve.

    An identity holds for all x, so there is nothing to solve — compare coefficients instead.

  • Comparing coefficients without expanding first.

    The sides must be in the same form before their terms can be matched.

  • Matching an x term with a constant.

    Compare like with like: x terms with x terms, constants with constants.

  • Working on both sides when proving an identity.

    Manipulate one side only until it becomes the other.

Exam tips

  • Look for the ≡ symbol to identify an identity.
  • Expand fully before comparing anything.
  • Set out the comparisons as separate labelled equations.
  • Substitute a value such as x = 1 to check your coefficients.

Key terms

Identity
An equation true for all values of the unknown.
Coefficient
The number multiplying a term.
Expand
To multiply out brackets.
Constant
A term with no variable.

Related topics

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