From what a variable is to solving equations with brackets — every core algebra skill explained clearly, with fully worked examples and the mistakes that cost students marks.
Algebra is the branch of maths that uses letters to represent numbers whose values are unknown or changing. Instead of writing "a number plus 5 equals 12," algebra lets you write x + 5 = 12 and then solve for x. This shorthand makes it possible to describe patterns and solve problems that would be impossible to express in plain words.
The letter used is called a variable or unknown. Any letter can be used, but x, y, n, and a are the most common at this level.
You can only add or subtract like terms. Think of it like adding apples and oranges — you cannot combine them into a single fruit type.
Example: Simplify 5x + 3y − 2x + 4y
Group the x terms: 5x − 2x = 3x
Group the y terms: 3y + 4y = 7y
Answer: 3x + 7y
Example: Simplify 4a² + 3a − a² + 2 − 5a
a² terms: 4a² − a² = 3a²
a terms: 3a − 5a = −2a
Constants: 2
Answer: 3a² − 2a + 2
To expand a bracket, multiply the term outside the bracket by every term inside it.
Single bracket: a(b + c) = ab + ac
Example: Expand 3(2x − 5)
3 × 2x = 6x; 3 × (−5) = −15
Answer: 6x − 15
Example with a negative outside: −2(4x + 3)
−2 × 4x = −8x; −2 × 3 = −6
Answer: −8x − 6
Multiplying a negative by a positive gives a negative. Multiplying two negatives gives a positive. Get these sign rules right and expanding brackets becomes straightforward.
To expand (a + b)(c + d), multiply every term in the first bracket by every term in the second. A reliable memory aid is FOIL: First, Outside, Inside, Last.
Example: Expand (x + 3)(x + 5)
First: x × x = x²
Outside: x × 5 = 5x
Inside: 3 × x = 3x
Last: 3 × 5 = 15
Collect like terms: x² + 5x + 3x + 15 = x² + 8x + 15
Example with a negative: Expand (x − 4)(x + 2)
x² + 2x − 4x − 8 = x² − 2x − 8
Factorising is the reverse of expanding. You look for the highest common factor (HCF) of all terms and take it outside a bracket.
Example: Factorise 12x + 8
HCF of 12 and 8 is 4. → 4(3x + 2)
Example: Factorise 6x² − 9x
HCF of 6x² and 9x is 3x. → 3x(2x − 3)
Always check your factorisation by expanding the bracket — you should get back to the original expression.
Solving an equation means finding the value of the unknown that makes both sides equal. The key rule: whatever you do to one side, do the same to the other. This keeps the equation balanced.
Work through these steps: simplify each side, collect all unknown terms on one side, collect all number terms on the other side, then divide to isolate the unknown.
Solve: x + 7 = 15
Subtract 7 from both sides: x = 15 − 7 = 8
Solve: 3x − 4 = 14
Add 4 to both sides: 3x = 18
Divide both sides by 3: x = 6
Solve: 5x + 2 = 2x + 14
Subtract 2x from both sides: 3x + 2 = 14
Subtract 2: 3x = 12
Divide by 3: x = 4
Solve: 2(3x − 1) = 4x + 10
Expand: 6x − 2 = 4x + 10
Subtract 4x: 2x − 2 = 10
Add 2: 2x = 12
Divide by 2: x = 6
Always expand brackets before trying to move terms across the equals sign. Skipping this step is the single most common error in algebra.
Substitution means replacing a variable with a given value to find the numerical result of an expression.
Question: Find the value of 3x² − 2x + 1 when x = 4.
= 3(4²) − 2(4) + 1
= 3(16) − 8 + 1
= 48 − 8 + 1
= 41
Work out powers first (BIDMAS/BODMAS), then multiply, then add and subtract.
Many exam questions describe a situation in words and ask you to write an algebraic expression or equation. Key phrases to recognise:
Question: Tickets cost £t each. Sara buys 4 tickets and receives £3 change from £20. Write an equation and solve it.
Equation: 4t + 3 = 20
4t = 17 → t = £4.25
Algebra is the language of all advanced maths. Every topic you meet later — simultaneous equations, quadratics, trigonometry, statistics — uses the skills on this page. Practise them until they feel automatic.