GCSE · Maths · AQA · Spec 8300 · Higher

Identities (Higher)

Any x makes 3(x + 2) = 3x + 6 true. No x makes 3(x + 2) = 3x + 8 true. One digit apart: what is the difference?

Algebra · What kind of statement?

Same sign, different claims

Sort the same eight statements three ways. Start with the sign. Then ask which values make each one true. Then name each one.

What sign does it use, if any?

Still to sort

= or ≡ (0)

An equals-type sign joins two sides.

<, >, ≤ or ≥ (0)

An inequality sign compares two sides.

No sign (0)

Nothing joins it to anything else.

8 of 8 still to sort.

Pick a statement, then the column it belongs in. Each placement shows the reason, right or wrong.

Showing an identity

How do you show it's true for every x?

You are told that 3(x + 2) ≡ 3x + 6: the two sides are equal for every value of x. You have to show that this is true.

Which is closest to what you would do?

Put in a few numbers. x = 1, x = 2 and x = 10 all work, so it's shown.Solve it for x, the same way as any equation.It's written with ≡, so it's already true. There's nothing to show.Expand the left-hand side and show that it becomes exactly the right-hand side.

How sure are you?

I'm sureI think soI'm guessing

Put in a few numbers. x = 1, x = 2 and x = 10 all work, so it's shown.

This is a common misconception

Those checks are worth doing, but they are evidence, not proof. Three values aren't every value: x could be 1000.5 or −⅓ or any number you will never try, and a statement can pass every check you make and still fail at one you didn't. Checking works in one direction only. A single value where the sides differ DISPROVES an identity: x = 0 turns 3(x + 2) = 3x + 8 into 6 = 8, which is false. No number of successful checks proves one.

Solve it for x, the same way as any equation.

This is a common misconception

Try it. Expanding gives 3x + 6 = 3x + 6, and taking 3x from both sides leaves 6 = 6, with no x anywhere. That isn't a mistake. The x's cancelled because the two sides were the same expression all along, and 6 = 6 is true whatever x was, so there is no particular answer to find. (Do the same to 3(x + 2) = 3x + 8 and you get 6 = 8, which is never true, so that equation has no solution.) Getting 6 = 6 tells you the statement is an identity. To SHOW it, write the argument the other way round: start from one side and turn it into the other.

It's written with ≡, so it's already true. There's nothing to show.

This is a common misconception

The ≡ sign is a claim, not a proof. Anyone can write 3(x + 2) ≡ 3x + 8, and it's false. The sign tells your reader what you are claiming; the algebra is what shows the claim is true.

Expand the left-hand side and show that it becomes exactly the right-hand side.

This matches the evidence

Yes. 3(x + 2) = 3 × x + 3 × 2 = 3x + 6, which is the right-hand side. Each step is true for every value of x, so the conclusion is too, and no value has been left out. That is what an algebraic argument gives you that checking numbers can't.

Examples can knock an identity down, but only algebra can show it is true. Start from one side and turn it into the other.

Maths · Algebra

Show that, one line at a time

Work on ONE side only. Step forward to reveal each line and the operation that produced it.

GoalShow that (n + 1)² − (n − 1)² ≡ 4n

LHS = (n + 1)² − (n − 1)²

start from the left

Write down the left-hand side only. Don't write '= 4n' yet: that is what you are trying to show.

Step 1 of 5

Write down the left-hand side only. Don't write '= 4n' yet: that is what you are trying to show.

Watch out: Don't write the whole statement at the top and then change both sides at once. That treats the claim as true before you have shown it. Transform one side until it matches the other.

Higher

Algebraic proof

Now write a proof yourself

Prove that the sum of any three consecutive integers is a multiple of 3. [4 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

Writes three consecutive integers in general form, e.g. n, n + 1 and n + 2 where n is any integer (or n − 1, n, n + 1). (1 mark)
Adds and simplifies correctly: n + (n + 1) + (n + 2) = 3n + 3 (or 3n from n − 1, n, n + 1). (1 mark)
Shows 3 as a factor: 3n + 3 = 3(n + 1). (1 mark)
Concludes in words: n + 1 is an integer, so 3(n + 1) is a multiple of 3 whatever integer n is (or: 3n is 3 times an integer). (1 mark)

Model answer

Let the three consecutive integers be n, n + 1 and n + 2, where n is any integer. Their sum is n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1). Since n + 1 is an integer, 3(n + 1) is a multiple of 3. So the sum of any three consecutive integers is a multiple of 3.

Choose letters that stand for every case: n, n + 1, n + 2 for consecutive integers, 2n for any even number, 2n + 1 for any odd number. Two numbers that are not linked need different letters, such as 2m + 1 and 2n + 1. Then simplify, pull out the factor you need, and end with a sentence saying what the algebra shows.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Some statements are true only for particular values of x. An identity is true for every value, and you show it with algebra, not examples.

What you need to know

Expressions, equations, formulae, inequalities and identities are told apart by what they claim, not by whether they contain an = sign.
An equation is true only for particular values of its letter, possibly none. An identity is true for every value, and can be written with ≡.
Terms are added or subtracted; factors are multiplied.
Trying values can show a statement is not an identity, but it can never prove that it is one.
To show two expressions are identical, start with one side and use algebra to turn it into the other.
Proof (Higher): write the general case with letters, such as n, n + 1 or 2n + 1, and finish with a sentence saying what the algebra shows.

The big picture

An equation is true only for particular values of its letter, and sometimes for none. An identity is true for every value and can be written with ≡. Whether a statement uses = or ≡ doesn't decide which it is; the values that make it true do. You show an identity by turning one side into the other with algebra. Trying values can prove a statement is not an identity, but never that it is one. A proof uses general letters such as n, n + 1 and 2n + 1 so that it covers every case at once.

Key points

1Some values → equation. Every value → identity (≡).

2The = sign alone decides nothing.

3Terms: added. Factors: multiplied.

4One counterexample disproves an identity; examples never prove one.

5Show that: one side → the other side, then conclude.

6Proof: general letters, simplify, factorise, concluding sentence.

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Worked example

Problem

5(x + a) + 3 ≡ 5x + 18. Find the value of the constant a.

⚠ Watch out

Calling a statement an identity because the values you tried worked. x² + 2 = 3x works for x = 1 and x = 2, but x = 0 gives 2 = 0: it is an equation. A check can rule an identity out, never in.

🧠

Memory hook

Some x → equation. Every x → identity. Examples can break an identity; only algebra can make one.

✓

Check yourself

Without substituting any numbers, decide whether 4(x − 3) + 12 = 4x is an equation or an identity, and show why in two lines.

Flashcards

(9)

Equation or identity: 5(x − 1) = 5x − 5?

Identity. The left side expands to exactly 5x − 5, so it is true for every x. It can be written 5(x − 1) ≡ 5x − 5.

What does the symbol ≡ mean?

'Is identically equal to': the two sides are equal for every value of the letter.

What is the difference between a term and a factor?

Terms are added or subtracted, like 4x and 12 in 4x + 12. Factors are multiplied, like 4 and x + 3 in 4(x + 3).

What is a formula?

A rule linking different quantities, such as C = 2πr for the circumference of a circle, so that one can be worked out from the others.

Is x² + 4 > 0 an identity?

No. It is true for every x, but it is an inequality. An identity says two expressions are equal for every value.

Why doesn't checking x = 1, 2 and 3 prove an identity?

Three values aren't every value, so it could still fail for one you didn't try. Checks can disprove an identity but never prove one.

You 'solve' a statement and are left with 7 = 7. What does that tell you?

The letter has cancelled and what is left is always true, so the statement holds for every value. It is an identity.

How do you set out a 'show that' for an identity?

Start from one side, change it step by step into the other side, then state that the two sides are identical.

In a proof, how do you write any even number and any odd number?

Even: 2n. Odd: 2n + 1. Here n stands for any integer.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Same sign, different claims

How do you show it's true for every x?

Show that, one line at a time

Now write a proof yourself

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