Algebraic vocabulary: knowing what each object actually is
Examiners regularly ask "is this an expression, an equation, a formula or an identity?" — usually for a single mark, but it's free if you know the definitions, and a guaranteed loss if you don't. Lock these down once and you keep the mark forever.
The five objects
Expression — a collection of terms with no equals sign.
Examples: 3x + 5, a² - 4ab, (x + 2)/y. You can simplify or evaluate an expression, but you cannot "solve" it because there's nothing to solve.
Equation — two expressions joined by =, true for specific values of the unknowns.
Example: 2x + 3 = 11 is true only when x = 4. You solve an equation.
Formula — an equation that expresses one quantity in terms of others, used as a rule.
Example: A = πr², v = u + at, C = 5(F − 32)/9. Each variable carries a meaning (Area, radius, etc.). You substitute into or rearrange a formula.
Identity — an equation that is true for all values of the unknown. Often written with the symbol ≡.
Example: 2(x + 3) ≡ 2x + 6. Both sides describe the same thing in different forms. Examiners use identities for proofs.
Inequality — like an equation but with <, >, ≤ or ≥.
Example: 3x - 1 < 11. The solution is a range (here x < 4), not a single value.
Other essential vocabulary
Term — a single number/letter or product, separated by + or −. In 5x² - 3x + 7 the terms are +5x², -3x, +7.
Coefficient — the numerical factor in a term. In -3x the coefficient is -3 (sign included).
Variable — a letter standing for an unknown quantity (x, y, t …).
Constant — a number on its own, with no letter (the +7 in the term list above).
Factor (algebraic) — an algebraic expression that divides exactly into another. (x + 2) is a factor of x² + 5x + 6 because x² + 5x + 6 = (x + 2)(x + 3).
✦Worked example— Worked example: classify
State whether each is an expression, equation, formula or identity.
- (a)
5x - 7 - (b)
5x - 7 = 18 - (c)
y = 5x - 7(where y is a variable depending on x) - (d)
(x + 3)² ≡ x² + 6x + 9
Answers: (a) expression; (b) equation; (c) formula (or "linear equation in two variables" — the exam mark scheme accepts either, but treats y = mx + c as a formula in this section); (d) identity.
⚠Common mistakes— Common mistakes (examiner traps)
- Calling everything an "equation". Expressions have no equals sign; formulae and identities have an equals sign but a different use. The mark is for the precise word.
- Confusing identity and equation. An identity holds for every x; an equation only for specific values. The
≡symbol is the giveaway. - Saying "solve a formula". You don't solve formulae — you rearrange or substitute into them.
- Mixing up coefficient and term. In
-3xthe coefficient is-3, the term is-3x. - Forgetting that "constant" requires no letter. In
5x + 2yneither term is a constant.
➜Try this— Quick check
Classify: P = 2(l + w) — formula. a² + 2ab + b² ≡ (a + b)² — identity. 2y + 5 = 17 — equation. 4y - 1 ≥ 11 — inequality. x² - x — expression.
AI-generated · claude-opus-4-7 · v3-deep-algebra