GCSE Maths · Algebra

Expressions, equations and identities GCSE Questions and Worked Answers

Q: What is the difference between = and ≡?

= states equality. ≡ emphasises that two expressions are equal for every allowed input, not just a solution of a particular equation.

Q: Do I always need to solve an equation?

Follow the question. You might only be asked to form it, identify it, or check whether a supplied value satisfies it.

An expression represents a value. An equation states that two values are equal; a formula relates quantities. An identity is an equality true for every allowed input. An inequality compares values using less-than or greater-than signs.

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Start with the meaning

What you need to know about expressions, equations and identities

Imagine buying three identical pens. If each costs £2, the total is 3 × 2 = £6. If you do not yet know the price, call it x pounds. The total is then 3 × x, usually written 3x. This is an expression: a way to describe a value without yet knowing its numerical amount.

See the idea first

The extra information changes what you can find

Now suppose the three pens cost £12 altogether. Writing 3x = 12 gives an equation: the amounts on the two sides are equal. Dividing both totals by 3 tells you that one pen costs £4, so x = 4. The expression 3x alone did not tell you that; the equation supplied a condition.

The same purchase, with different information
Written statement	Meaning	What you can do
3x	Cost of three pens	Evaluate it if x is supplied
3x = 12	The total is £12	Solve to find x = 4
C = 3x	C is the total cost in pounds	Use a formula for different prices
3x ≤ 12	The total is at most £12	Find all permitted prices

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Write and evaluate an expression

What the problem asks: A notebook costs £x and a folder costs £2. Write the cost of two notebooks and one folder, then find it when x = 3.

How to solve it: The expression is 2x + 2. Evaluate means find its numerical value: replace x with 3, giving 2 × 3 + 2 = 8, so £8.

Form and solve an equation

What the problem asks: Two notebooks at £x each and a £2 folder cost £10. Find the notebook price.

How to solve it: The condition gives 2x + 2 = 10. Subtract 2 from both sides, then divide both sides by 2: x = 4. Doing the same operation to equal amounts preserves equality.

Use a formula

What the problem asks: The cost C pounds of n tickets at £5 each is given by C = 5n. Find the cost of four tickets.

How to solve it: A formula is an equation used to relate quantities. Here n counts tickets and C gives their total cost. Substituting n = 4 gives C = 20, or £20. A formula is not a completely separate category from an equation.

Decide whether an equality is an identity

What the problem asks: Is 2(x + 3) = 2x + 6 true for every x, or only selected values?

How to solve it: Multiplying each part inside the bracket by 2 gives 2x + 6 for every x. This is an identity. The symbol ≡ emphasises ‘equal for every allowed input’: 2(x + 3) ≡ 2x + 6. A few numerical checks alone cannot establish an identity.

Describe a limit using an inequality

What the problem asks: A bag can hold at most 8 kg. Write a statement for its load m kilograms.

How to solve it: Write m ≤ 8. The symbol ≤ means less than or equal to, so 8 is allowed. By contrast, m < 8 excludes 8. The symbols > and ≥ mean greater than, and greater than or equal to.

A reliable routine

Choose the action the question actually asks for

These words describe different mathematical objects, not one universal calculation. Once you understand the statement, use the instruction to decide what result is needed. An identity and a formula are special uses of equalities; the categories can overlap.

Write an expression when asked to represent an amount.
Evaluate by substituting supplied values. Simplify by rewriting the same value more compactly, such as 3x + 2x = 5x.
Solve an equation by finding the values that make its equality true. For an inequality, find the allowed range or set of values.
To establish an identity, show with valid algebra why both expressions agree for every allowed input, not just one test value.

Check: A term is a part added or subtracted in an expression: 3x + 2 has terms 3x and 2. A factor is multiplied: 3 and x are factors of 3x. ‘Allowed input’ matters when division is involved: a denominator cannot be zero.

Fully worked

Expressions, equations and identities GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

An expression is not an equation

2 marks

Question

Write an expression for five more than twice n. Evaluate it for n = 6.

Twice n means 2n; five more gives $2n+5$ . Now use the supplied value.

2(6)+5

=12+5

=17

Tip: do not add ‘= 0’ to the expression; no such condition was given.

Example 2

An equation supplies a condition

3 marks

Question

Five more than twice a number is 19. Find the number.

Call the number n.

2n+5=19

Subtract 5 from both equal sides.

2n=14

Divide both sides by 2.

n=7

Check: twice 7 plus 5 is 19.

Example 3

A formula connects quantities

3 marks

Question

A taxi charges £4 plus £3 per kilometre. Write a formula for cost C pounds for d kilometres. Find C when d = 6.

The distance charge is 3d pounds; add the fixed £4.

C=4+3d

Substitute d = 6.

C=4+3(6)

C=22

The fare is £22. Tip: define both letters and their units.

Example 4

An identity holds for every input

3 marks

Question

Show that $4(x+2)\equiv4x+8$ . Explain how this differs from $4x+8=20$ .

Distribute 4 to both terms inside the bracket.

4(x+2)=4x+4(2)

=4x+8

This works for every real x, so it is an identity. In contrast:

4x+8=20

4x=12

x=3

That equation is true only when x = 3.

Example 5

A limit is not a single value

3 marks

Question

Tickets cost £4 each. You can spend at most £18. Write an inequality for the number n of tickets and find the maximum whole number you can buy.

The total is 4n, and ‘at most’ includes equality.

4n\le18

Divide by positive 4, preserving the inequality direction.

n\le4.5

Tickets come in whole numbers, so the maximum is 4. Check: four cost £16; five cost £20.

Example 6

One matching value is not an identity

2 marks

Question

Someone claims $x^2\equiv2x$ because both sides are 4 when x = 2. Is this a valid identity?

No. Try the allowed input x = 3.

x^2=9

2x=6

The two values differ. This one counterexample disproves the identity. Matching at x = 2 only shows that 2 solves the equation $x^2=2x$ .

10 original questions · total 19 marks

Expressions, equations and identities GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 24 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

Represent an amount

1 mark

Write an expression for the cost of four tickets at £t each and a £3 booking fee.

Show worked answer

Four tickets cost 4t pounds. Add the one booking fee: $4t+3$ .

Evaluate

2 marks

Evaluate $4t+3$ when t = 5.

Show worked answer

4(5)+3

=20+3

=23

Replace the letter before calculating.

Simplify

1 mark

Simplify $6a+2a$ . Do you now know a?

Show worked answer

Six lots of a plus two lots is $8a$ . No value of a is determined: this is a simpler expression, not an equation to solve.

Form an equation

3 marks

Four tickets at £t each plus a £3 fee cost £31. Find t.

Show worked answer

4t+3=31

4t=28

t=7

Each ticket costs £7. Check: 4 × £7 + £3 = £31.

Use a formula

2 marks

A rectangle has perimeter P = 2l + 2w, where all lengths are in cm. Find P when l = 9 and w = 4.

Show worked answer

P=2(9)+2(4)

P=18+8

P=26

The perimeter is 26 cm; P is the quantity calculated by this formula.

Identify an identity

2 marks

Which holds for every real x: $3(x+4)=3x+12$ or $3x+12=27$ ? Explain.

Show worked answer

Expanding $3(x+4)$ gives $3x+12$ for any x, so the first is an identity. The second gives $3x=15$ , hence x = 5 only.

Read an inequality

2 marks

Write an inequality for a temperature T strictly below 6°C. Is 6°C included?

Show worked answer

$T<6$ . Strictly below excludes the boundary, so 6°C is not included. Use ≤ only if equality is allowed.

Terms and factors

2 marks

In $7x+5$ , name the two terms and two factors of the term 7x.

Show worked answer

The terms are 7x and 5: they are added. The factors 7 and x are multiplied to make 7x.

Overlapping descriptions

2 marks

A square's perimeter is P = 4s. Explain why this is both an equation and a formula.

Show worked answer

It is an equation because it states equality. It is used as a formula because it relates perimeter P and side length s, measured in the same length units. The two labels do not have to be mutually exclusive.

Disprove an identity

2 marks

Disprove the claim $(x+1)^2\equiv x^2+1$ .

Show worked answer

Choose x = 1.

(1+1)^2=4

1^2+1=2

These differ, so the proposed equality is not an identity. Checking x = 0 would not disprove it because that particular input happens to work.

Examiner-style feedback

Common expressions, equations and identities mistakes

Solving an expression

3x + 5 does not give a condition to solve. Only evaluate it if an input is supplied, or simplify it if asked.

Treating the labels as separate boxes

A formula is an equation relating quantities. An identity states equality for every allowed input.

Proving by a few substitutions

Agreement at selected inputs does not establish an identity. Use algebra for all inputs; one failing input is enough to disprove it.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

An expression represents a value.
An equation states equality; a formula relates quantities.
An identity holds for every allowed input.
An inequality describes a comparison, including or excluding its boundary.

Quick answers

Expressions, equations and identities FAQ

What is the difference between = and ≡?

= states equality. ≡ emphasises that two expressions are equal for every allowed input, not just a solution of a particular equation.

Do I always need to solve an equation?

Follow the question. You might only be asked to form it, identify it, or check whether a supplied value satisfies it.

Build connected skills

What to revise next

Use supplied values

Substitution

Evaluate expressions and formulae step by step.

Revise substitution

Find unknowns

Linear equations

Keep both sides equal while isolating an unknown.

Revise linear equations

Rewrite equal expressions

Expanding brackets

See why a bracket can be replaced by its expansion.

Revise expanding brackets

Content standards

Curriculum and rights review

A1–A3 algebraic vocabulary across tiers: expressions, equations, formulae, identities, inequalities, terms and factors. Simple identities here prepare for, but do not replace, Higher formal algebraic proof. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content