Checking equivalent expressions | Year 9

01 / LEARN

The idea, step by step

One matching input is not enough

Equivalent expressions give the same value for every input allowed by both forms. $4 (x + 2)$ and $4 x + 8$ are equivalent because the distributive law connects them.

$2 x$ and $x + 3$ both give 6 when $x = 3$ , but give 2 and 4 when $x = 1$ . Agreement at one input proves only agreement there, not equivalence for all inputs.

A matching value can be a coincidence
x	2x	x + 3
1	2	4
3	6	6
5	10	8

Prove or disprove appropriately

To prove an equivalence, transform one expression into the other using valid algebra, or simplify both to the same form. Explain the operations rather than joining unrelated expressions with equals signs.

One permitted input giving different values is enough to disprove equivalence. This is a counterexample. A finite table can suggest an identity, but cannot by itself establish equality for every real input. Check any denominator restrictions before choosing a test value.

A common trap

Treating one successful numerical test as proof of an identity.

Instead: An algebraic argument supports a universal claim; a counterexample can disprove it.

02 / WORKED EXAMPLES

See the thinking

Example 1

Prove equivalence

Are $3 (2 x - 1) + 4$ and $6 x + 1$ equivalent?

Expand: $3 (2 x - 1) + 4 = 6 x - 3 + 4$ .
Collect constants: $6 x + 1$ .
This equals the second expression for every real $x$ .

Yes; both simplify to

6 x + 1

Example 2

Find a counterexample

Test the claim that $5 (x - 2)$ and $5 x - 2$ are equivalent.

Choose the permitted input $x = 0$ .
The first expression gives $- 10$ and the second gives $- 2$ .
They differ, so the claim is false. The correct expansion is $5 x - 10$ .

Not equivalent;

x = 0

is a counterexample.

Check your understanding

Are $2 (x + 5) - x$ and $x + 10$ equivalent?

Show answer

Yes.

$2 (x + 5) - x = 2 x + 10 - x = x + 10$ for every real $x$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Are $4 x + 7 x$ and $11 x$ equivalent? Explain.
2 marks
Are $3 (x + 2)$ and $3 x + 2$ equivalent? Give a counterexample if not.
2 marks
Show $5 a - 2 (a - 3)$ is equivalent to $3 a + 6$ .
2 marks
Are $6 x + 9$ and $3 (2 x + 3)$ equivalent? Explain.
2 marks
Two expressions agree at $x = 2$ . Explain why this does not prove equivalence, using your own pair.
3 marks
Show $\frac{8 x}{4}$ and $2 x$ are equivalent. Is $x = 0$ allowed?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Are $4 x + 7 x$ and $11 x$ equivalent? Explain.
2 marks
$4 x + 7 x = (4 + 7) x = 11 x$ for all $x$ .
Yes.
Are $3 (x + 2)$ and $3 x + 2$ equivalent? Give a counterexample if not.
2 marks
$3 (0 + 2) = 6$ but $3 (0) + 2 = 2$ .
No; $x = 0$ gives 6 and 2.
Show $5 a - 2 (a - 3)$ is equivalent to $3 a + 6$ .
2 marks
$5 a - 2 (a - 3) = 5 a - 2 a + 6 = 3 a + 6$ .
Both simplify to $3 a + 6$ .
Are $6 x + 9$ and $3 (2 x + 3)$ equivalent? Explain.
2 marks
Expanding the second expression gives $6 x + 9$ .
Yes.
Two expressions agree at $x = 2$ . Explain why this does not prove equivalence, using your own pair.
3 marks
At $x = 2$ both give 4.
At $x = 0$ the values are 2 and 0, so they are not equivalent.
For example, $x + 2$ and $2 x$ agree at 2 but differ at 0.
Show $\frac{8 x}{4}$ and $2 x$ are equivalent. Is $x = 0$ allowed?
3 marks
$\frac{8 x}{4} = 2 x$ because 4 is non-zero.
The denominator does not contain $x$ .
They are equivalent, including at $x = 0$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Year 9 algebraic simplification and equivalent expressions, with consolidation of linear formula skills. Binomial expansion and quadratic factorisation/equations are in the separate Quadratics and Parabolas chapter.

Official curriculum source ↗

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