What does it mean to 'simplify' an expression?

Combine like terms — terms with the same letter (and power). 2x + 5 + 3x − 2 = 5x + 3. The 2x and 3x combine, the 5 and -2 combine.

KS3 Algebra Foundations: The Skills You'll Use Forever

Q: What algebra is taught in KS3?

KS3 algebra (Years 7–9) covers using letters for unknowns, substitution, simplifying expressions, expanding and factorising brackets, solving linear equations, sequences, and an introduction to graphs.

Q: What's the difference between an expression and an equation?

An expression is a maths phrase without an equals sign (e.g. 3x + 5). An equation contains an equals sign and can be solved for a value (e.g. 3x + 5 = 14 → x = 3).

Algebra is using letters to represent unknown or changing numbers. KS3 lays the foundations for everything that comes after — get these five skills solid by end of Year 9 and GCSE algebra becomes manageable. This is the map.

The five foundational skills

#	Skill	Year typically introduced
1	Using letters for unknowns + substitution	7
2	Simplifying expressions (collecting like terms)	7
3	Expanding brackets	7–8
4	Factorising	8
5	Solving linear equations	7–9

Every GCSE algebra topic — quadratics, simultaneous equations, inequalities, graphs — builds on these. Master them in KS3 and Year 10 starts easy.

1. Using letters and substitution

A letter (like x, y, a) stands for a number. The number can be:

Unknown (in an equation, you solve to find it)
Variable (in a formula, it changes with the problem)

Substitution example

If y = 3x + 2, find y when x = 4:

y = 3(4) + 2
y = 12 + 2
y = 14

The brackets around (4) aren't strictly needed but they make multiplication explicit. Useful habit.

2. Simplifying expressions

Combine like terms — terms with the same letter and same power.

2x + 5 + 3x − 2 = 5x + 3

Why? 2x + 3x = 5x (like terms). 5 − 2 = 3 (like terms, constants). The x term and the constant don't combine.

Less obvious case

3a + 2b − a + 5b = 2a + 7b

3a − a = 2a. 2b + 5b = 7b. The a terms don't combine with b terms.

Watch the signs

4x − 7 − 2x + 3 = 2x − 4

4x − 2x = 2x. −7 + 3 = −4.

See mastering negative numbers for sign rules.

3. Expanding brackets

The distributive property: multiply outside by each inside term.

3(x + 4) = 3x + 12
2(2y − 5) = 4y − 10
-3(x − 1) = -3x + 3      (watch the double negative)

Full guide: Expanding brackets in KS3.

4. Factorising (the reverse of expanding)

Start with 3x + 12. Notice both terms share a factor of 3. Pull it out:

3x + 12 = 3(x + 4)

Process

Find the highest common factor (HCF) of the terms
Divide each term by the HCF
Write as HCF × bracket of remaining terms

More examples

6a + 9 = 3(2a + 3)        (HCF is 3)
4x − 8 = 4(x − 2)         (HCF is 4)
5xy + 10x = 5x(y + 2)     (HCF is 5x — both terms share x)

Verify

Always check by expanding: does 3(2a + 3) = 6a + 9? Yes ✓.

5. Solving linear equations

Find the value of x that makes both sides equal. Do the same to both sides until x is alone.

3x + 5 = 14
3x = 9       (subtract 5 from both sides)
x = 3        (divide by 3)

Full guide: Solving linear equations in KS3.

How these five connect

A typical KS3 algebra problem chains them:

"Solve 2(x + 3) = 14"

Expand the bracket: 2x + 6 = 14
Subtract 6 from both sides: 2x = 8
Divide by 2: x = 4

Three of the five skills in three lines. The whole point of mastering each individually is so you can chain them fluently.

Notation conventions

3x means 3 × x (multiplication is implied)
x² means x × x (x squared)
2x² means 2 × x × x (NOT (2x)² = 4x²)
(x + 3)² means (x + 3) × (x + 3) (NOT x² + 9!)

The squaring trap (line 4 above) catches many KS3 students. Always expand the bracket first.

Common KS3 algebra mistakes

Mistake	Wrong	Right
Partial expansion	`3(x + 4) = 3x + 4`	`3x + 12`
Sign error in expansion	`−2(x − 3) = −2x − 6`	`−2x + 6`
Squaring a sum	`(x + 3)² = x² + 9`	`(x + 3)² = x² + 6x + 9`
Forgetting both sides	`3x = 15 → x = 15`	`x = 5`
Like-term confusion	`2x + 3 = 5x`	`2x + 3` (can't combine)

See 5 maths mistakes Year 8 students make for fixes.