Free GCSE Maths lesson: Algebra
Aplailasain by Static Carnival

Free Lessons → GCSE / Key Stage 4 → Maths → Algebra Notation and Substitution

Lesson 26 · GCSE / Key Stage 4 · Maths · Algebra

Algebra Notation and Substitution

Learn how algebra is written, what each part of an expression means, and how to substitute values accurately.

Qualification: GCSE Key Stage 4 Subject: Maths Strand: Algebra

GCSE specification fit

A foundation 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 guidanceFoundation and Higher

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.

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
Parts of an algebraic expression The expression 3a plus 2b minus 5 is labelled to show terms, coefficients, variables and the constant term. 3a + 2b − 5 term term constant 3 is a coefficient a and b are variables

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 × a ab means a × b 4(x + 2) means 4 × (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 × 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 × 4² − 3 = 2 × 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
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
Answer: 14.

Example 3: Use powers after substitution

Find 3x² + 2 when x = 5.

3x² + 2 = 3 × 5² + 2 = 3 × 25 + 2 = 75 + 2
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
Answer: £28.

Quick checks

Choose an answer, then check your thinking.

1. What does 6p mean?

2. Which one is an equation?

3. Find 2x² + 1 when x = 3.

Practice questions

Question 1

In the expression 8m + 3, state the coefficient of m.

Reveal answer and marking guidance

Answer: 8.

Marking: 8m means 8 × m, so the coefficient of m is 8.

Question 2

Is 5y − 2 an expression or an equation?

Reveal answer and marking guidance

Answer: expression.

Marking: It has no equals sign, so it is an expression.

Question 3

Find 4a + 7 when a = 6.

Reveal answer and marking guidance

Answer: 31.

Marking: 4a + 7 = 4 × 6 + 7 = 24 + 7 = 31.

Question 4

Find 2p − 3q when p = 9 and q = 4.

Reveal answer and marking guidance

Answer: 6.

Marking: 2 × 9 − 3 × 4 = 18 − 12 = 6.

Question 5

Find x² + 5x when x = −2.

Reveal answer and marking guidance

Answer: −6.

Marking: (−2)² + 5 × (−2) = 4 − 10 = −6.

Question 6

A phone plan costs C = 12 + 4g pounds, where g is the number of extra gigabytes used. Find the cost when g = 3.

Reveal answer and marking guidance

Answer: £24.

Marking: C = 12 + 4 × 3 = 24. The answer is a cost, so use pounds.

Question 7

Find 6 − 2x² when x = −3.

Reveal answer and marking guidance

Answer: −12.

Marking: Use brackets for the negative value: 6 − 2 × (−3)² = 6 − 2 × 9 = 6 − 18 = −12.

Question 8

The formula A = bh gives the area of a rectangle in cm². Find A when b = 7.5 cm and h = 4 cm.

Reveal answer and marking guidance

Answer: 30 cm².

Marking: A = 7.5 × 4 = 30. The formula gives area, so the unit is cm².

Question 9

Find 5t² − 3t + 8 when t = −2.

Reveal answer and marking guidance

Answer: 34.

Marking: Use brackets for the negative value: 5 × (−2)² − 3 × (−2) + 8 = 20 + 6 + 8 = 34.

Question 10

The formula P = 2l + 2w gives the perimeter of a rectangle in cm. Find P when l = 8.5 and w = 3.

Reveal answer and marking guidance

Answer: 23 cm.

Marking: Substitute both values: P = 2 × 8.5 + 2 × 3 = 17 + 6 = 23. The formula gives a length, so the unit is cm.

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For algebra notation and substitution, marks usually come from reading terms and coefficients correctly, replacing each letter with the given value, using brackets around negative substitutions, and simplifying in the correct order before adding units if the formula represents a context.

Common mistakes

  • Reading 3a as 3 + a: 3a means 3 × a.
  • Confusing expressions and equations: an equation has an equals sign; an expression does not.
  • Squaring too late or too early: in 2x², square x first, then multiply by 2.
  • Dropping brackets with negatives: if x = −4, write (−4) when substituting.
  • Ignoring the context: if the formula gives a length, cost or time, include the correct unit in the answer.

Extension challenge

Find 3a² − 2ab when a = −4 and b = 5.

Reveal answer

Answer: 88.

Use brackets: 3 × (−4)² − 2 × (−4) × 5 = 3 × 16 − (−40) = 48 + 40 = 88.

Exam-board guidance

Algebra notation and substitution questions can test vocabulary, formulae, negative values, powers and units. The safest routine is to identify each letter, substitute with brackets where needed, then simplify in the correct order.

AQA GCSE Maths

Name coefficients and expressions accurately, then use brackets for negative substitutions and show powers before multiplication or addition.

OCR GCSE Maths

Translate notation such as 5a and a² first, then substitute values in the stated order and check whether the final value fits the context.

Pearson Edexcel GCSE Maths

Show the full substitution line with each letter replaced, especially when a formula has two variables, powers or a real-life unit.

Eduqas GCSE Maths

Keep multiplication signs, powers and brackets clear in the working, then attach the correct unit when the formula represents a practical quantity.

WJEC Wales

In numeracy-style formulae, check that each substituted value matches the named variable and that the final answer has the right unit.

CCEA GCSE Maths

Use brackets during substitution, simplify in the correct order, and practise the routine for both calculator and non-calculator unit questions.

Next lesson

Next, move into Simplifying Expressions.

Previous lesson: Growth, Decay and Iteration Problems Next lesson: Simplifying Expressions