Brackets and variables on both sides | Year 8

01 / LEARN

The idea, step by step

Distribute before collecting

A factor outside a bracket multiplies every term inside. Thus $3 (x - 2) = 3 x - 6$ , not $3 x - 2$ . Expanding makes the variable terms and constants easier to compare.

Subtract a variable term from both sides to gather the unknown on one side. Subtracting the smaller coefficient often leaves a positive one, although either valid choice works.

Choose a useful route

Sometimes dividing first is shorter: $4 (x + 1) = 20$ becomes $x + 1 = 5$ . When both sides contain variables, expanding first often helps.

Check negative values using brackets. The goal is not a memorised instruction to move a term; it is a chain of statements with exactly the same solution.

A common trap

Expanding $4 (x - 2)$ to $4 x - 2$ .

Instead: The 4 multiplies both x and negative 2, giving $4 x - 8$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

One bracket

Solve $3 (x + 2) = 2 x + 11$ .

Expand: $3 x + 6 = 2 x + 11$ .
Subtract $2 x$ and then 6.
Both original sides equal 21 at x = 5.

x = 5

Example 2

Two brackets

Solve $2 (x - 3) = 3 (x - 5)$ .

Expand: $2 x - 6 = 3 x - 15$ .
Subtract $2 x$ and add 15: $9 = x$ .
The original sides both equal 12.

x = 9

Check your understanding

Solve $2 (x + 4) = x + 13$ .

Show answer

$x = 5$ .

Expand, subtract x and then subtract 8.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Solve $2 (x + 3) = 18$ .
2 marks
Solve $3 (x - 1) = 2 x + 5$ .
2 marks
Solve $5 (x + 2) = 3 x + 18$ .
2 marks
Solve $4 (x - 2) = 2 (x + 3)$ .
2 marks
Explain and correct the expansion $2 (x + 5) = 2 x + 5$ .
2 marks
Plan A costs AUD 8 plus AUD 3 per hour; B costs AUD 2 plus AUD 5 per hour. When are they equal?
3 marks

Read the worked solutions

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

Solve $2 (x + 3) = 18$ .
2 marks
Divide by 2 to get $x + 3 = 9$ .
Subtract 3.
$x = 6$ .
Solve $3 (x - 1) = 2 x + 5$ .
2 marks
Expand: $3 x - 3 = 2 x + 5$ .
Subtract $2 x$ and add 3.
$x = 8$ .
Solve $5 (x + 2) = 3 x + 18$ .
2 marks
Expand, then subtract $3 x$ and 10: $2 x = 8$ .
Divide by 2.
$x = 4$ .
Solve $4 (x - 2) = 2 (x + 3)$ .
2 marks
Expand to $4 x - 8 = 2 x + 6$ .
Subtract $2 x$ , add 8 and divide by 2.
$x = 7$ .
Explain and correct the expansion $2 (x + 5) = 2 x + 5$ .
2 marks
The outside factor acts on every term.
Both x and 5 must be doubled.
The correct expansion is $2 x + 10$ .
Plan A costs AUD 8 plus AUD 3 per hour; B costs AUD 2 plus AUD 5 per hour. When are they equal?
3 marks
Set $8 + 3 h = 2 + 5 h$ .
Subtract $3 h$ and 2: $6 = 2 h$ , so h = 3.
At 3 hours, both cost AUD 17.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics (2026)

Multi-step equations; brackets and variables on both sides; equations with rational solutions; substitution checks; one-variable inequalities and number-line solutions; tables and straight-line graphs; horizontal/vertical lines; how coefficients change a graph; graphical solutions and intersection checks; real-world linear models with digital exploration.

Official curriculum source ↗

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