Equations vs Identities and Algebraic Arguments Revision Notes | GCSE AQA Higher Maths

1. The Core Distinction: Equations vs Identities

Here is a question that trips up a surprising number of students: is $2x + 6 = 10$ the same kind of statement as $2(x + 3) \equiv 2x + 6$? They both involve algebra and an equals sign — but they are fundamentally different objects, and the mark schemes treats them differently.

An equation is a puzzle: it is only true for certain values of the variable. When you solve an equation, you are hunting for those specific values. An identity, by contrast, is always true — no matter what value you substitute in, both sides give the same result.

The clearest way to see the difference is to test a statement with two different values. Take $3x - 1 = 8$: substitute $x = 3$ and you get $8 = 8$ — true. Substitute $x = 5$ and you get $14 = 8$ — false. That is the hallmark of an equation: it fails for most values.

Now test $3(x + 2) \equiv 3x + 6$ with $x = 5$: left side gives $3 \times 7 = 21$, right side gives $15 + 6 = 21$. Try $x = -4$: left gives $3 \times (-2) = -6$, right gives $-12 + 6 = -6$. It works every time — because expanding the bracket on the left always produces the right-hand side. That universality is what makes it an identity.

2. Recognising and Verifying Identities

Knowing the definition is the starting point. The exam will ask you to decide whether a given statement is an equation or an identity — and then justify your answer algebraically. The method is always the same: expand and simplify both sides, then compare.

If both sides simplify to the same expression, the statement is an identity. If they do not match, it is an equation (or possibly a false statement altogether). This approach works on both calculator-allowed questions and questions without a calculator, since it relies on algebraic manipulation rather than arithmetic.

Worked example: Deciding: Equation or Identity?

Decide whether $4(x + 3) - 2x = 2(x + 6)$ is an equation or an identity. Justify your answer.

1. $4(x + 3) - 2x = 4x + 12 - 2x$ — Multiply each term inside the bracket by 4, then write the $-2x$ term alongside. [M1 for correct expansion]
2. $4x + 12 - 2x = 2x + 12$ — Collect like terms: $4x - 2x = 2x$, constant stays as $12$.
3. $2(x + 6) = 2x + 12$ — Multiply each term inside the bracket by 2.
4. $\text{LHS} = 2x + 12 \quad \text{RHS} = 2x + 12$ — Both sides are identical expressions — they match for every value of $x$. [A1 for correct conclusion]
5. $4(x + 3) - 2x \equiv 2(x + 6)$ — Because LHS $=$ RHS for all values of $x$, this is an identity. Write $\equiv$ in the final statement.

Answer: This is an identity: $4(x + 3) - 2x \equiv 2(x + 6)$, since both sides simplify to $2x + 12$ for all values of $x$.

3. Constructing Algebraic Arguments

Now the lesson moves into territory that separates the top grades from the rest. An algebraic argument is a structured chain of algebraic steps that proves a mathematical claim is true in general — not just for one or two numbers you happened to try.

The key word here is general. If a question asks you to show that the sum of two consecutive even numbers is always even, testing $2 + 4 = 6$ and noticing $6 \div 2 = 3$ tells you something useful. You need algebra to cover every possible case at once.

The standard approach is to represent unknown numbers using algebraic expressions, then manipulate those expressions to reach the required conclusion. The table below shows how to encode common number types algebraically — these representations appear repeatedly in GCSE Higher exam questions.

With these representations in hand, you can build an argument that covers every integer at once. The worked example below shows the full structure the examiner expects — notice how each step follows logically from the last, and the conclusion is stated explicitly at the end.

Worked example: Algebraic Argument: Sum of Two Consecutive Odd Numbers

Show that the sum of any two consecutive odd numbers is always a multiple of 4.

1. $\text{First odd number: } 2n + 1$ $\text{Second odd number: } 2n + 3$ — Let $n$ be any integer. Consecutive odd numbers differ by 2, so the next one is $(2n+1) + 2 = 2n + 3$. [M1 for correct algebraic representation]
2. $(2n + 1) + (2n + 3)$ — Add the two expressions together.
3. $= 2n + 1 + 2n + 3 = 4n + 4$ — Collect the $n$ terms: $2n + 2n = 4n$. Collect the constants: $1 + 3 = 4$. [M1 for correct simplification]
4. $4n + 4 = 4(n + 1)$ — Factor out 4. This shows the result is 4 multiplied by an integer $(n+1)$. [A1 for factorised form]
5. $4(n+1)$ is a multiple of 4 for all integer values of $n$. — The conclusion must be stated explicitly — do not leave the examiner to infer it. [A1 for conclusion]

Answer: The sum of any two consecutive odd numbers is $4(n+1)$, which is always a multiple of 4.

4. Showing Algebraic Expressions Are Equivalent

Earlier you saw how to verify an identity by expanding both sides. The same technique applies when a question asks you to show that two expressions are equivalent — the difference is that the target result is given to you, and your job is to reach it through valid steps.

The golden rule for 'show that' questions: start from what you know and work towards the given result. Never start from the answer and work backwards — that approach assumes what you are trying to prove, which earns no marks.

Worked example: Show That: Equivalent Expressions

Show that $(x + 3)^2 - (x + 1)^2 \equiv 4x + 8$.

1. $(x + 3)^2 = (x+3)(x+3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9$ — Expand by multiplying each term in the first bracket by each term in the second. [M1 for correct expansion of either squared bracket]
2. $(x + 1)^2 = (x+1)(x+1) = x^2 + x + x + 1 = x^2 + 2x + 1$ — Same method applied to the second bracket.
3. $(x^2 + 6x + 9) - (x^2 + 2x + 1)$ — Write the subtraction clearly. The brackets around the second expansion remind you to subtract every term inside it. [M1 for correct subtraction structure]
4. $= x^2 + 6x + 9 - x^2 - 2x - 1 = 4x + 8$ — $x^2 - x^2 = 0$; $6x - 2x = 4x$; $9 - 1 = 8$. The $x^2$ terms cancel. [A1 for reaching $4x + 8$]
5. $\therefore (x+3)^2 - (x+1)^2 \equiv 4x + 8$ as required. — Write 'as required' or 'QED' to signal the argument is complete. This is expected in 'show that' questions.

Answer: $(x+3)^2 - (x+1)^2 \equiv 4x + 8$, shown by expanding both squares and simplifying.

5. Practice: Putting It Together

The best way to consolidate these ideas is to work through problems that mix both skills: identifying the type of statement and constructing the argument. The practice set below covers the range of question styles you will meet in the Higher-tier exam.

6. Exam Technique and Common Mistakes

The ideas in this lesson are not difficult to understand — but they are easy to lose marks on through careless notation or incomplete conclusions. The guidance below targets the specific errors that appear most often in mark schemes.