Algebra: Expressions & Equations

A term is a single number, letter, or combination - like 3x, 5y², or 7. An expression is a collection of terms joined by + or −, like 3x + 5y − 2. It has no equals sign. An equation has an equals sign and can be solved - like 3x + 5 = 14. A formula is an equation that links variables, like A = lw (area = length × width). Knowing which you are dealing with tells you what you can and cannot do: you can simplify an expression but you cannot 'solve' it.

Like terms have exactly the same letter(s) raised to the same power. You can add or subtract like terms but not unlike terms. For example: 3x + 5x = 8x (both are x terms). But 3x + 5x² cannot be simplified - x and x² are different. In an expression like 4x + 3y − x + 2y, collect the x terms (4x − x = 3x) and the y terms (3y + 2y = 5y) separately to get 3x + 5y. Also remember: multiplying terms uses index laws - x² × x³ = x⁵ (add the powers). Dividing: x⁵ ÷ x² = x³ (subtract the powers).

Expanding means removing brackets by multiplying everything inside by the term outside. For a single bracket: 3(2x + 5) = 6x + 15. For two brackets (double expansion), use FOIL - First, Outer, Inner, Last. For (x + 3)(x + 5): First = x², Outer = 5x, Inner = 3x, Last = 15. Adding: x² + 5x + 3x + 15 = x² + 8x + 15. A special case to memorise: (a + b)² = a² + 2ab + b². Many students forget the middle term (2ab) - this is one of the most common errors in GCSE algebra.

A linear equation has no powers higher than 1. To solve it, do the same operation to both sides until x is alone. Work in reverse order of operations: deal with + and − first, then × and ÷. Example: 3x + 5 = 14. Subtract 5 from both sides: 3x = 9. Divide both sides by 3: x = 3. For equations with brackets, expand first: 2(3x − 1) = 16 → 6x − 2 = 16 → 6x = 18 → x = 3. For equations with unknowns on both sides: collect x terms on one side first. 5x + 3 = 2x + 12 → 3x = 9 → x = 3.

Factorising is the reverse of expanding - you put the expression back into brackets. For a simple expression, find the highest common factor (HCF) of all terms. For example: 6x² + 9x. The HCF of 6x² and 9x is 3x. So 6x² + 9x = 3x(2x + 3). Check by expanding: 3x × 2x = 6x² ✓, 3x × 3 = 9x ✓. To factorise a quadratic like x² + 7x + 12, find two numbers that multiply to 12 and add to 7. Those are 3 and 4. So x² + 7x + 12 = (x + 3)(x + 4). This skill is essential for solving quadratic equations.