A1–A3Algebraic notation, substitution and identities
Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Algebra
Practise Algebraic notation, substitution and identities. 1 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
How to write and read algebra correctly, substitute numbers into expressions and formulae, and use the words term, expression, equation, formula, identity and inequality. It comes up as quick 1 to 3 mark questions on both tiers and underpins every other algebra topic.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
1
Collect like termsAdd or subtract terms with exactly the same letters, e.g. \(3a + 5b - a + 2b = 2a + 7b\).
2
Write products in algebraic notationWrite \(a \times b\) as \(ab\), \(y \times y\) as \(y^2\) and \(3 \times a \times a \times b\) as \(3a^2b\).
3
Substitute positive numbers into expressionsReplace each letter by its value, e.g. when \(a = 4\) and \(b = 3\), \(5a - 2b = 20 - 6 = 14\).
3
Tell expressions, equations, formulae and identities apartAn expression has no equals sign, an equation can be solved, a formula links quantities and an identity is true for every value.
4
Substitute negative numbers into formulaePut negative values in brackets, e.g. when \(x = -3\), \(x^2 = (-3)^2 = 9\).
5
Use the identity symbol correctlyWrite \(\equiv\) when both sides are equal for every value of the letter, e.g. \(2(x + 3) \equiv 2x + 6\).
Notes
Writing algebra
Leave out the × sign and write the number first: \(a \times b = ab\) and \(b \times 3 = 3b\).
\(y + y + y = 3y\), but \(y \times y \times y = y^3\).
Division is written as a fraction: \(a \div b = \frac{a}{b}\). Write coefficients as fractions, not decimals: \(\frac{1}{2}x\) or \(\frac{x}{2}\), not \(0.5x\).
\(2a^2\) means \(2 \times a \times a\): only the a is squared. \((2a)^2 = 4a^2\).
Like terms have exactly the same letters and powers: \(3xy\) and \(-5yx\) are like terms, but \(x^2\) and \(x\) are not.
The vocabulary
Term: a number, a letter, or numbers and letters multiplied together, such as \(5\), \(4x\) or \(-3ab^2\).
Expression: one or more terms with no equals sign, e.g. \(4x - 7\).
Equation: has an equals sign and is true only for particular values, e.g. \(4x - 7 = 5\) is true only when \(x = 3\).
Formula: a rule linking two or more quantities, e.g. \(A = lw\). You substitute values into it.
Identity: true for every value of the letter, written with \(\equiv\), e.g. \(3(x + 2) \equiv 3x + 6\).
Inequality: uses \(\lt\), \(\gt\), \(\le\) or \(\ge\), e.g. \(2x + 1 \gt 9\).
Factor: something that divides exactly into a term, e.g. 3 and x are factors of \(3x\).
Substitution
Write the expression, replace each letter with its value in brackets, then use the order of operations (brackets, indices, then × and ÷, then + and −).
Put negative numbers in brackets: if \(t = -2\), then \(t^2 = (-2)^2 = 4\), but \(-t^2 = -(-2)^2 = -4\).
Scientific formulae work the same way: in \(v = u + at\) with \(u = 3\), \(a = 2\) and \(t = 5\), \(v = 3 + 2 \times 5 = 13\).
Identities
To show that an identity is true, expand and simplify one side until it is exactly the same as the other side.
Cheatsheet
\(ab = a \times b\) and \(\frac{a}{b} = a \div b\)
\(a^2 = a \times a\), \(a^3 = a \times a \times a\), \(a^2b = a \times a \times b\)
Like terms: same letters with the same powers
Expression: no equals sign. Equation: can be solved. Formula: links quantities. Identity (\(\equiv\)): true for all values
Substituting negatives: use brackets, \((-3)^2 = 9\)
Order of operations: brackets, indices, × and ÷, + and −
How to answer each type of question
Simplify an expression
1 to 2 marks2
Look at each term together with the sign in front of it.
Group like terms (same letters and powers).
Add or subtract the coefficients and write the answer with no like terms left.
Example. Simplify \(5p + 3q - 2p - 7q + p\)
Show the model answer
\(5p - 2p + p = 4p\) and \(3q - 7q = -4q\) \(4p - 4q\) B2 (B1 for \(4p\) or \(-4q\))
Choose the expression, equation or formula
1 mark each3
An expression has no equals sign.
An equation is true only for certain values; a formula links different quantities.
An identity uses \(\equiv\) and is true for all values.
Example. Here are four statements: \(5x - 2 = 13\), \(v = u + at\), \(4(y + 1) \equiv 4y + 4\), \(3p^2 - p\). From the list, write down (a) an expression (b) an equation (c) a formula.