Expressions, Equations and IdentitiesEdexcel GCSE Maths: Revision notes
Section 1
How Is Algebra Written Down?
Algebra uses shorthand notation that you need to read fluently before you can manipulate it.
- ab means a × b (the multiplication sign is dropped)
- 3y means y + y + y (three lots of y), not 3 to the power of y
- a² means a × a
- a/b means a ÷ b
- Coefficients can be fractions, e.g. (1/2)x means half of x
- Brackets group terms that must be treated as a single quantity, e.g. 2(x + 3) means the whole of (x + 3) is doubled
Reading notation correctly is essential — misreading 3y as 3 × y × y (instead of 3 × y) is a very common error.
Do not confuse 3y (= y + y + y) with y³ (= y × y × y) — the position of the number matters.
Section 2
Expression, Equation, Formula or Identity — Which Is It?
These four words are used precisely in GCSE Maths and examiners expect you to use them correctly.
| Word | What it is | Example |
|---|---|---|
| Expression | A collection of terms with no equals sign | 3x + 2 |
| Equation | A statement that two expressions are equal for specific value(s) of the variable | 3x + 2 = 11 |
| Formula | An equation linking two or more variables, used to calculate one from the others | A = πr² |
| Identity | A statement that is true for every value of the variable, shown with the symbol ≡ | x² − 1 ≡ (x − 1)(x + 1) |
An equation is only true for particular value(s) of x (it can be solved). An identity is true for all values of x (it doesn't need solving — both sides are always equal).
5x + 3 = 18 is an equation (only true when x = 3). x + x + x ≡ 3x is an identity (true for every value of x).
Section 3
Simplifying Expressions and Substituting Into Formulae
Simplifying by collecting like terms: like terms have exactly the same variable(s) raised to the same power. Add or subtract their coefficients, keep unlike terms separate.
Example: 5x + 3y − 2x + y = 3x + 4y
Substituting means replacing letters with numbers and evaluating. Always substitute into the whole term, including any powers or brackets, before doing arithmetic.
Example: if a = 4 and b = −2, evaluate a² + 3b: 4² + 3(−2) = 16 − 6 = 10
This also works with scientific formulae, e.g. substituting values into v = u + at to find a speed.
When substituting a negative number into a squared term, always use brackets: (−2)² = 4, but −2² is often misread as −4.
Section 4
Proving Two Expressions Are Equivalent
To show that two expressions are equivalent (i.e. they form an identity), expand, simplify or factorise one side until it matches the other exactly.
Steps:
- Start from the more complicated side
- Expand any brackets
- Collect like terms
- Compare with the target expression — if identical, the equivalence is proved
Example: show that 2(x+3) + x ≡ 3x + 6. 2(x+3) + x = 2x + 6 + x = 3x + 6 — matches the right-hand side, so the identity is proved.
Never move terms across the ≡ sign as if solving an equation — instead, simplify each side independently until they match.
Must Know
- An expression has no equals sign; an equation is true for specific value(s); a formula links variables; an identity (≡) is true for all values
- 3y means y + y + y (not y × y × y); ab means a × b; a² means a × a
- Collect like terms by adding/subtracting coefficients — unlike terms cannot be combined
- Substitute carefully into powers and brackets, especially with negative numbers
- To prove an identity, expand and simplify one side until it matches the other exactly
- Formulae and expressions can be evaluated by substitution, including scientific formulae
That's the notes covered.
Carry on to the next subtopic.