Equivalent linear expressions | Year 8

01 / LEARN

Understand the idea.

Equivalent for every allowed input

Equivalent expressions give the same output for every allowed input. The forms $2 (x + 3)$ and $2 x + 6$ are equivalent because distribution transforms the first into the second without choosing a particular x value.

To show equivalence, expand any brackets and collect like terms. If one form becomes the other through valid algebraic transformations, the argument covers all inputs. Different forms can help different tasks: brackets show groups, while collected terms reveal coefficients.

A match is not a proof

One matching input is not enough. At $x = 1$ , both $2 x + 1$ and $3 x$ give $3$ , but at $x = 2$ they give $5$ and $6$ . The first agreement was only a coincidence.

To disprove equivalence, one counterexample is enough: choose the same allowed input for both expressions and show different outputs. To prove equivalence, use an identity valid for all allowed inputs rather than relying on a finite list of substitutions.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Show two forms agree

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

Worked solution

1
Expand the bracket.
$4 (x + 2) - x = 4 x + 8 - x$
Distribution multiplies both x and two by four, without restricting x.
2
Collect the x terms.
$4 x + 8 - x = (4 - 1) x + 8 = 3 x + 8$
The coefficient of the subtracted x is negative one.
3
State what the identity establishes.
$4 (x + 2) - x = 3 x + 8 for every x$
Every transformation was valid for any x, so the expressions are equivalent.

Answer

Equivalent for every $x$ .

EXAMPLE 02

Find a counterexample

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

Worked solution

1
Choose one allowed input for both expressions.
$x = 0$
A single unequal output pair is sufficient to disprove equivalence.
2
Evaluate both expressions at that input.
$\begin{aligned} 2 (0) + 4 & = 4 \\ 4 (0) + 2 & = 2 \end{aligned}$
Using the same input makes the two outputs directly comparable.
3
Compare the outputs.
$4 \neq 2$
The expressions disagree at x equal to zero, so they are not equivalent for every x.

Answer

No; $x = 0$ is a counterexample.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Yes.

Yes. Distributing and collecting gives $3 (x + 2) + x = 3 x + 6 + x = 4 x + 6$ . These algebraic steps are valid for every x, so they establish equivalence.

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Equivalent for every allowed input

A match is not a proof

See the method in action.

Show two forms agree

Expand the bracket.

Collect the x terms.

State what the identity establishes.

Find a counterexample

Choose one allowed input for both expressions.

Evaluate both expressions at that input.

Compare the outputs.

Try it yourself.

Check your understanding

Keep practising