Algebra Notation and Substitution | GCSE Maths

Lesson overview

A starting point for every later algebra topic.

Algebra notation lets you write general rules clearly. Substitution lets you use those rules with actual numbers, including formulae from real contexts.

QualificationGCSE Mathematics

Key stageKey Stage 4

StrandAlgebra

Tier Both

Calculator status Calculator and non-calculator

Exam-board status Shared algebra skill

What you will learn

What variables, coefficients and terms mean.
How to read expressions and equations.
How multiplication and powers are written in algebra.
How to substitute values into expressions and formulae.
How to use the order of operations after substitution, especially with powers and negative values.
How to check whether an answer makes sense in context, including units and the meaning of each variable.

Key facts before you start

Variable A letter represents a value that may change or be unknown.

Coefficient The number multiplying a variable, such as 5 in 5x.

Substitution Replace the letter with a given value, then calculate carefully.

Notation habit 5x means 5 × x; avoid writing unnecessary multiplication signs in algebra.

Algebra notation and substitution infographic

Before viewing

Before viewing, identify a variable, a coefficient and one substitution step in the visual.

Infographic explaining GCSE Maths algebra notation and substitution, including variables, coefficients, terms, expressions, equations, brackets, powers, negative values and unit checks. — Use this visual to read algebra notation, substitute values safely and check order of operations, signs and units.Download visual

After viewing

After viewing, substitute x = 4 into a simple expression and show the arithmetic step.

Why this matters

Algebra is a compact way of describing patterns and relationships. It appears in sequences, graphs, formulae, geometry, proportion and problem solving.

Most mistakes at this stage come from notation rather than difficult maths. If you can read the notation accurately, later algebra becomes much more manageable.

Prior knowledge

You should already be comfortable with:

adding, subtracting, multiplying and dividing numbers,
using the order of operations,
working with powers such as 4²,
using negative numbers carefully,
reading a question and identifying the quantities involved.

Clear explanation

Variables, coefficients and terms

A variable is a letter that stands for a number. In GCSE Maths, letters such as a, b, n, x and y are often used as variables.

A coefficient is the number multiplying a variable. A term is one part of an expression, separated by + or − signs.

Expression: 3a + 2b - 5 Terms: 3a, 2b and -5 Coefficient of a: 3 Coefficient of b: 2

Expressions and equations

An expression is a mathematical phrase. It does not have an equals sign.

4x + 7 is an expression

An equation says two things are equal. It has an equals sign and can often be solved.

4x + 7 = 31 is an equation

Multiplication conventions

In algebra, multiplication is usually written without the × sign. This avoids confusion when x is being used as a variable.

3a means 3 \times a ab means a \times b 4(x + 2) means 4 \times (x + 2)

Write numbers before letters, and write letters in alphabetical order where it is sensible: 5ab is clearer than b × 5 × a.

Powers in algebra

A power applies to the thing directly before it. In a², the letter a is multiplied by itself.

a² = a × a 3a² means 3 × a², not (3a)² (3a)² means 3a × 3a

Substitution

To substitute means to replace a letter with a number. A good habit is to rewrite the expression with brackets first, then simplify.

Find 3a + 4 when a = 5 3a + 4 = 3 \times 5 + 4 = 15 + 4 = 19

Order of operations after substitution

After substitution, use the normal order of operations: brackets, powers, multiplication and division, then addition and subtraction.

Find 2x² - 3 when x = 4 2x² - 3 = 2 \times 4² - 3 = 2 \times 16 - 3 = 32 - 3 = 29

Brackets are especially useful with negative numbers.

If x = −3, then x² = (−3)² = 9 but −3² means −(3²) = −9

Checking with units and contexts

If a formula comes from a real context, keep the units in mind. For example, if C = 2n + 5 gives a cost in pounds and n is the number of items, the final answer should be money.

n = 6 gives C = 2 × 6 + 5 = 17, so the cost is £17

Worked examples

Example 1: Identify parts of an expression

In the expression 7x − 4, name the coefficient of x and the constant term.

7x means 7 × x, so the coefficient of x is 7 −4 is the constant term

Reveal answer

Answer: coefficient 7, constant term −4.

Example 2: Substitute into an expression

Find 5a − 2b when a = 4 and b = 3.

5a − 2b = 5 × 4 − 2 × 3 = 20 − 6

Reveal answer

Answer: 14.

Example 3: Use powers after substitution

Find 3x² + 2 when x = 5.

3x² + 2 = 3 × 5² + 2 = 3 × 25 + 2 = 75 + 2

Reveal answer

Answer: 77.

Example 4: Substitute into a formula with units

A taxi fare is modelled by F = 3d + 4, where F is the fare in pounds and d is the distance in miles. Find the fare for 8 miles.

F = 3 × 8 + 4 = 24 + 4

Reveal answer

Answer: £28.

Quick checks

Choose an answer, then check your thinking.

1. What does 6p mean?

6 + p 6 × p 6 − p

2. Which one is an equation?

4x + 9 2a − 5b 3n + 1 = 16

3. Find 2x² + 1 when x = 3.

19 37 13