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Foundations · Algebra

Algebra basics: variables, expressions, expanding and factorising

Letters as numbers, collecting like terms, substituting values, expanding brackets with the area model, and taking out a common factor.

First, the rules

A letter is a number you have not been told yet

x behaves exactly like 7 or 12: you can add copies of it, multiply it, put it in brackets. Every algebra rule is an arithmetic rule with the number hidden.

Notation
Example 1
3x means 3 × x; xy means x × y; x² means x · x; x4 means x ÷ 4. A term is one product; an expression is terms added.
★ Like terms
Example 1
Same letters, same powers: 4x + 7x = 11x, but 4x + 7y and 4x + 7x² cannot be combined.
★ Substitution
Example 1
Replace the letter, then follow the order of operations. x = −3: 2x² − 5x = 2·9 + 15 = 33 (brackets around negatives!).
★ Expanding a single bracket
Example 2
a(b + c) = ab + ac — 3(2x − 5) = 6x − 15; −2(x − 4) = −2x + 8.
★ Expanding two brackets
Example 2
(x + a)(x + b) = x² + (a + b)x + ab — four products (area model), then collect the middle.
★ Factorising (common factor)
Example 3
ab + ac = a(b + c) — take out the HCF of the terms: 6x² + 9x = 3x(2x + 3). Expand to check.

See it

Expanding as an area

A rectangle x + a wide and x + b tall. Its four pieces are the four products of the expansion; change a, b, and x itself, and the numeric check always balances.

Step by step

Worked examples

1

Collect, then substitute

like terms · substitution

Simplify 5a + 3b − 2a + 7 − b and evaluate it for a = 4, b = −2.

Ask yourself: Group the a terms, the b terms and the plain numbers.

Show the solution
  1. a terms: 5a − 2a = 3a. b terms: 3b − b = 2b. Numbers: 7. So 3a + 2b + 7.
  2. Substitute: 3(4) + 2(−2) + 7 = 12 − 4 + 7 = 15.
  3. Check in the original: 20 − 6 − 8 + 7 + 2 = 15 ✓.

Answer 3a + 2b + 7; 15

2

Expanding brackets

expanding

Expand and simplify (a) 4(2x − 3) − 2(x + 5), (b) (x + 6)(x − 2).

Ask yourself: Multiply everything inside by what is outside — including the sign.

Show the solution
  1. (a) 8x − 12 − 2x − 10 = 6x − 22.
  2. (b) x·x + x·(−2) + 6·x + 6·(−2) = x² − 2x + 6x − 12 = x² + 4x − 12.

Answer 6x − 22 · x² + 4x − 12

3

Taking out the common factor

factorising

Factorise (a) 12x + 18, (b) 10x² − 15x, (c) 4ab + 6a².

Ask yourself: What is the biggest number — and the biggest letter power — in every term?

Show the solution
  1. (a) HCF 6: 6(2x + 3).
  2. (b) HCF 5x: 5x(2x − 3).
  3. (c) HCF 2a: 2a(2b + 3a). Expand each one back to check.

Answer 6(2x + 3) · 5x(2x − 3) · 2a(2b + 3a)

Practice: algebra basics

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Like terms, substitution, expanding single and double brackets, and taking out a common factor. Your best score is kept in this browser only.

About this lesson

Algebra basics for school mathematics: what a variable is, simplifying by collecting like terms, substituting numbers into an expression, expanding single and double brackets with the area model, and factorising by taking out the highest common factor. An area model you can move, worked examples and practice questions with explanations. Everything on this page runs in your browser; nothing you type is sent or stored. Spotted an error? Tell us.