Equivalent linear expressions | Year 8

01 / LEARN

The idea, step by step

Equivalent for every allowed input

Expressions are equivalent if they have the same value for every allowed input. The distributive property makes $2 (x + 3)$ and $2 x + 6$ equivalent. We can use either form without changing the relationship.

Expand brackets and collect like terms in both expressions. If both simplify to the same form, the steps justify equivalence. A bracketed form can show groups while an expanded form makes coefficients easier to compare.

A match is not a proof

The expressions $2 x + 1$ and $3 x$ both equal $3$ at $x = 1$ . At $x = 2$ , however, they give $5$ and $6$ , so they are not equivalent. One permitted input with different outputs is a counterexample.

A single matching input can be a coincidence. To establish an identity, use transformations valid for all inputs. To disprove one, state the input clearly and show both resulting values.

A common trap

Calling two expressions equivalent after checking just one input.

Instead: Use valid algebra to show agreement for every input; one matching value cannot establish that.

02 / WORKED EXAMPLES

See the thinking

Example 1

Show two forms agree

Show that $4 (x + 2) - x$ and $3 x + 8$ are equivalent.

Expand to $4 x + 8 - x$ .
Collect to get $3 x + 8$ .
The two forms are identical.

Equivalent for every

x

Example 2

Find a counterexample

Are $2 x + 4$ and $4 x + 2$ equivalent?

Use $x = 0$ .
The outputs are $4$ and $2$ .
One unequal pair disproves equivalence.

No;

x = 0

is a counterexample.

Check your understanding

Is $3 (x + 2) + x$ equivalent to $4 x + 6$ ?

Show answer

Yes.

Expansion and collection give $3 x + 6 + x = 4 x + 6$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Show that $2 (x + 5)$ and $2 x + 10$ are equivalent.
2 marks
Show that $5 x + 3 x - 4$ and $8 x - 4$ are equivalent.
2 marks
Is $3 (x - 2)$ equivalent to $3 x - 2$ ? Give a counterexample.
2 marks
Simplify $6 (x + 1) - 2 x$ and compare with $4 x + 6$ .
2 marks
Why is checking $2 x + 1$ and $3 x$ only at $x = 1$ not enough?
3 marks
Give an expanded expression and a factorised expression equivalent to $12 x + 8$ .
3 marks

Read the worked solutions

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

Show that $2 (x + 5)$ and $2 x + 10$ are equivalent.
2 marks
Distribute $2$ to get $2 x + 10$ .
Equivalent.
Show that $5 x + 3 x - 4$ and $8 x - 4$ are equivalent.
2 marks
Collect $5 x + 3 x = 8 x$ .
Equivalent.
Is $3 (x - 2)$ equivalent to $3 x - 2$ ? Give a counterexample.
2 marks
Evaluate both at the same permitted input.
No; at $x = 0$ the outputs are $- 6$ and $- 2$ .
Simplify $6 (x + 1) - 2 x$ and compare with $4 x + 6$ .
2 marks
Expand to $6 x + 6 - 2 x$ .
Collect to get $4 x + 6$ .
They are equivalent.
Why is checking $2 x + 1$ and $3 x$ only at $x = 1$ not enough?
3 marks
Equivalence means all allowed inputs, not just one.
They match there but differ at $x = 2$ , where the values are $5$ and $6$ .
Give an expanded expression and a factorised expression equivalent to $12 x + 8$ .
3 marks
Split or collect like terms without changing the value.
Extract common factor $4$ .
For example, $8 x + 4 x + 8$ and $4 (3 x + 2)$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 Mathematics with SA curriculum context

Combining like terms; substitution with signed/rational values; multiplying/dividing algebraic terms; expanding a single bracket; factorising common factors; equivalent linear expressions; rearranging simple formulas.

Official curriculum source ↗

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