Equations vs identities vs formulae; algebraic equivalence

A1 introduced the vocabulary; A6 takes it further into proof and argument. OCR Higher Paper 4 sets "show that" questions where you must verify an identity.

Quick recap

• Equation: holds for specific values. e.g. 2x + 3 = 11 ⇒ x = 4.
• Identity (≡): holds for ALL values. e.g. (x + 1)² ≡ x² + 2x + 1.
• Formula: defines one variable in terms of others. e.g. A = πr².

Showing an identity

Two methods:

Method 1 — Manipulate one side to match the other. Show LHS expand/simplify = RHS.

Example: Show that (x + 3)² − (x − 3)² ≡ 12x.

• LHS = (x² + 6x + 9) − (x² − 6x + 9)
• = x² + 6x + 9 − x² + 6x − 9
• = 12x = RHS ✓.

Method 2 — Manipulate both sides to a common form. Show LHS and RHS both reduce to the same expression.

"Show that" — algebraic argument

OCR Higher mark schemes credit method marks for each step.

• M1 for expanding brackets correctly.
• M1 for collecting like terms.
• A1 for the final equivalence.

Don't skip steps! "Show that" means show every step. Going LHS = RHS in two leaps loses M1 method marks.

Verifying versus solving

• "Show that 2(x + 5) = 2x + 10" — this is showing an IDENTITY (true for all x).
• "Solve 2(x + 5) = 36" — this is solving an EQUATION (find specific x).

Identities you should know

• (a + b)² ≡ a² + 2ab + b².
• (a − b)² ≡ a² − 2ab + b².
• (a + b)(a − b) ≡ a² − b² (difference of two squares).
• a(b + c) ≡ ab + ac (distributive).

Algebraic equivalence vs equality

• "Equivalent" expressions reduce to the same form (e.g. 2(x + 3) and 2x + 6 are equivalent).
• Two equations can be equivalent if they have the same solution set (e.g. 2x = 6 and x = 3 are equivalent equations).

OCR mark scheme conventions

• "Show that" → method marks for each correct step.
• The use of ≡ vs = is rarely penalised at GCSE but ≡ is preferred for identities.
• "Verify by substitution" — substitute a numerical value to check (NOT a proof, but useful for checking your algebra).