Simplify expressions by collecting like terms, expand single and double brackets, factorise into brackets, substitute into expressions and formulae, and change the subject of a formula.

What this dot point is asking

Algebraic manipulation is the toolkit the rest of algebra is built from: collecting like terms, expanding single and double brackets, factorising back into brackets, substituting numbers into expressions and formulae, and changing the subject of a formula. WJEC tests these directly and embeds them inside equations, sequences and graphs, so fluency here protects marks everywhere. The skills appear on both components, with double-bracket expansion, harder factorising and rearranging formulae with the subject appearing more than once reserved for the higher demand.

Collecting like terms

Like terms have identical letter parts, including powers, so $5x$ and $2x$ are like, but $5x$ and $5x^2$ are not. Add or subtract the coefficients only, leaving the letter part unchanged: $7a + 3b - 2a + b = 5a + 4b$. The letters are just labels keeping different quantities apart, which is why you can only combine matching ones.

Expanding brackets

Expanding removes brackets by multiplying out.

For a single bracket, multiply every term inside by the term outside: $4(3x - 2) = 12x - 8$. Watch the signs, since a negative outside flips every sign inside: $-2(x - 5) = -2x + 10$.

A common special case is the difference of two squares: $(x + a)(x - a) = x^2 - a^2$, where the middle terms cancel.

Factorising

Factorising is the reverse of expanding: write an expression as a product.

The first move is always to take out the highest common factor of all terms: $6x^2 + 9x = 3x(2x + 3)$, because $3x$ divides both terms. You can check a factorisation by expanding it back. At Higher tier you also factorise quadratics into double brackets, which is covered in the quadratic equations dot point.

Substituting into formulae

To substitute, replace each letter with its given value, using brackets to keep signs and powers safe, then evaluate with BIDMAS. For $v = u + at$ with $u = 3$, $a = -2$, $t = 4$: $v = 3 + (-2)(4) = 3 - 8 = -5$. Brackets around a negative value prevent sign slips, especially with squares: $(-3)^2 = 9$, not $-9$.

Changing the subject of a formula

Rearranging a formula uses the same inverse-operation logic as solving an equation, treating the other letters as numbers.

Work in reverse BIDMAS order, undoing the operations around the wanted letter and doing the same to both sides. To make $t$ the subject of $v = u + at$: subtract $u$ to get $v - u = at$, then divide by $a$ to get $t = \dfrac{v - u}{a}$. When the subject appears inside a power or root, undo that operation last, as in making $r$ the subject of $A = \pi r^2$.

Why this matters

These manipulations are the grammar of algebra. Every equation you solve, every sequence rule you find and every graph you analyse rests on expanding, factorising and rearranging cleanly. Because WJEC awards method marks for correct algebraic steps even when the final line slips, setting out each line of working clearly is worth real marks, and the non-calculator Unit 1 rewards confident, accurate manipulation.