Checking equivalent expressions | Year 9

01 / LEARN

Understand the idea.

One matching input is not enough

Two expressions are equivalent when they give equal values for every permitted input. An algebraic law can establish this for all inputs at once: expanding $4 (x + 2)$ gives $4 x + 8$ regardless of the real value of $x$ .

One matching input does not establish equivalence. The expressions $2 x$ and $x + 3$ both give $6$ at $x = 3$ , but at $x = 1$ they give $2$ and $4$ . They agree sometimes without being equivalent.

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 equivalence, transform one expression into the other using valid algebra, or simplify both to the same form. Each equality must join expressions with the same value for all permitted inputs, rather than merely report a convenient numerical test.

To disprove equivalence, find one counterexample: a permitted input for which the values differ. One failure is sufficient; many matching tests are still not a proof of an identity. Check denominator restrictions before choosing a test input.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Prove equivalence

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

Worked solution

1
Expand the first expression.
$3 (2 x - 1) + 4 = 6 x - 3 + 4$
The distributive law is valid for every real x.
2
Collect the constant terms.
$6 x - 3 + 4 = 6 x + 1$
Adding -3 and 4 gives 1 without changing the x term.
3
Compare with the second expression.
$3 (2 x - 1) + 4 = 6 x + 1 for all real x$
The algebra reaches exactly the second form, so the expressions are equivalent.

Answer

Yes; both simplify to $6 x + 1$ .

EXAMPLE 02

Find a counterexample

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

Worked solution

1
Choose a permitted test input.
$x = 0$
Both expressions are defined at zero, making it a valid candidate counterexample.
2
Evaluate the first expression.
$5 (0 - 2) = - 10$
The entire difference inside the brackets is multiplied by 5.
3
Evaluate the second expression.
$5 (0) - 2 = - 2$
Only x is multiplied by 5 in this expression.
4
Use the unequal values to reject the claim.
$- 10 \neq - 2$
One permitted input with unequal outputs disproves equivalence. Expanding correctly gives $5 x - 10$ .

Answer

Not equivalent; $x = 0$ is a counterexample.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Yes.

Yes. Expanding and collecting gives $2 (x + 5) - x = 2 x + 10 - x = x + 10$ for every real $x$ , which proves equivalence.

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

One matching input is not enough

Prove or disprove appropriately

See the method in action.

Prove equivalence

Expand the first expression.

Collect the constant terms.

Compare with the second expression.

Find a counterexample

Choose a permitted test input.

Evaluate the first expression.

Evaluate the second expression.

Use the unequal values to reject the claim.

Try it yourself.

Check your understanding

Keep practising