Simple equivalent expressions | Year 7

01 / LEARN

The idea, step by step

Count copies of the same quantity

Equivalent expressions have the same value for every allowed value of their variables. The terms $3 x + 2 x$ count three copies and two more copies of the same quantity. Together they are $5 x$ . Add their coefficients, not the letters.

The terms $3 x$ and 2 are unlike: one varies with $x$ and the other stays fixed. Keep $3 x + 2$ as two terms. Similarly, $3 x + 2 y$ cannot be turned into $5 x y$ , because it adds different quantities rather than multiplying them.

Explain a general rule

Three groups of $x + 2$ contain three copies of $x$ and six extra units. Therefore $3 (x + 2) = 3 x + 6$ for every $x$ . This explanation uses the structure of the groups. Also, $4 b + b = 5 b$ ; we normally write $b$ , not $1 b$ .

Substitution is useful for finding a false rule: one counterexample is enough. Equal results for one chosen value do not prove a rule for every value. To prove a simple equivalence, explain the grouping or operation that works generally.

A common trap

Combining $5 x + 3$ into $8 x$ .

Instead: The 3 is not three copies of $x$ . Keep the unlike terms separate.

02 / WORKED EXAMPLES

See the thinking

Example 1

Combine lengths

A path has sections $4 x$ metres and $3 x$ metres long. Give one expression for the total.

Both terms count copies of $x$ .
Four copies plus three copies make seven copies.

7 x

metres.

Example 2

Test a claim

Leah says $2 (x + 3) = 2 x + 3$ for every $x$ . Check $x = 4$ .

Left side: $2 (4 + 3) = 14$ .
Right side: $2 \times 4 + 3 = 11$ .
The results differ. The correct expansion is $2 x + 6$ .

The claim is false.

Check your understanding

Simplify $6 m + m$ .

Show answer

$7 m$

Six copies and one more copy make seven copies of $m$ .

03 / WORKSHEET

Your turn

6 questions · 9 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Simplify $2 x + 5 x$ .
1 mark
Simplify $8 a - 3 a$ .
1 mark
Write $p + p + p + p$ more briefly.
1 mark
Simplify $3 b + 2 + b + 5$ .
2 marks
Write $2 (n + 4)$ without brackets.
2 marks
Is $4 x + 2$ equivalent to $6 x$ ? Use $x = 3$ to explain.
2 marks

Read the worked solutions

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

Simplify $2 x + 5 x$ .
1 mark
Add the coefficients: $2 + 5 = 7$ .
$7 x$
Simplify $8 a - 3 a$ .
1 mark
Eight copies minus three leave five copies.
$5 a$
Write $p + p + p + p$ more briefly.
1 mark
There are four identical terms.
$4 p$
Simplify $3 b + 2 + b + 5$ .
2 marks
$3 b + b = 4 b$ .
$2 + 5 = 7$ .
$4 b + 7$
Write $2 (n + 4)$ without brackets.
2 marks
There are two copies of $n$ and two copies of 4.
$2 n + 8$
Is $4 x + 2$ equivalent to $6 x$ ? Use $x = 3$ to explain.
2 marks
$4 \times 3 + 2 = 14$ .
$6 \times 3 = 18$ .
A counterexample disproves equivalence.
No: the values are 14 and 18.

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Curriculum reference & lesson scope

Australian Curriculum v9.0 / SA Curriculum Mathematics

Year 7 variables, expression notation, substitution, simple equivalence and formula tables. Digital methods are explained with supplied data; the topic test works by hand.

Official curriculum source ↗

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