Algebraic Equivalence

Algebraic Equivalence

GCSE(F), GCSE(H),

It is often necessary to rearrange expressions to identify solutions to equations and to provide information when plotting graphs.

An equivalent expression, or identity, is shown with ≡ (an equals sign with an additional line).

To show that a given equivalence is an identity, expand, factorise or simplify one side to match the other side.

The two sides of the equivalence are known as the Left Hand Side (LHS) and the Right Hand Side (RHS), referring to the left and right side of the equals or equivalence sign.

Examples

1. Show that x2 ≡ (x + 3)(x - 3) + 9

Answer: RHS: x2 + 3x - 3x - 9 + 9 = x2

Therefore x2 ≡ (x + 3)(x - 3) + 9

2. Show that (x + 4)(x + 2) ≡ (x + 3)(x + 3) - 1.

Answer: RHS: (x + 4)(x + 2)

= x2 + 4x + 2x + 8

= x2 + 6x + 8.

LHS: (x + 3)(x + 3) - 1

= x2 + 3x + 3x + 9 - 1

= x2 + 6x + 8.