Equations with rational solutions | Year 8

01 / LEARN

The idea, step by step

Fractions are valid answers

A solution such as $3 / 2$ or $- 2 / 5$ is just as valid as an integer. For example, $2 x = 3$ gives exactly $x = 3 / 2$ . Rounding to 2 changes the solution.

A fraction bar groups its numerator. Read $(x + 1) / 3$ as one-third of the whole expression x + 1.

Clear denominators without losing terms

Multiplying both sides by a non-zero common denominator can remove fractions. Multiply every term, including constants outside a fraction.

Decimal coefficients can be scaled similarly by 10 or 100. Whatever route you choose, check the exact answer in the original statement.

A common trap

Rounding a fractional solution before checking it.

Instead: Keep the fraction exactly, or use an equal terminating decimal.

02 / WORKED EXAMPLES

See the thinking

Example 1

A rational solution

Solve $4 x + 1 = 7$ .

Subtract 1: $4 x = 6$ .
Divide by 4 and simplify $6 / 4$ to $3 / 2$ .
Check $4 (3 / 2) + 1 = 7$ .

x = 3 / 2

Example 2

A grouped numerator

Solve $(x + 2) / 3 = 5 / 2$ .

Multiply both sides by 6: $2 (x + 2) = 15$ .
Expand and solve $2 x + 4 = 15$ .
Check $(11 / 2 + 2) / 3 = 5 / 2$ .

x = 11 / 2

Check your understanding

Solve $3 x = 5$ .

Show answer

$x = 5 / 3$ .

Divide by 3; multiplication by 3 checks the exact result.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Solve $2 x = 7$ .
2 marks
Solve $5 x + 2 = 4$ .
2 marks
Solve $3 x - 1 = - 3$ .
2 marks
Solve $(x + 1) / 4 = 3 / 2$ .
2 marks
Solve $0.5 x + 1.2 = 3.7$ .
2 marks
Solve $x / 2 + x / 3 = 5$ .
3 marks

Read the worked solutions

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

Solve $2 x = 7$ .
2 marks
Divide both sides by 2.
Check $2 (7 / 2) = 7$ .
$x = 7 / 2$ .
Solve $5 x + 2 = 4$ .
2 marks
Subtract 2 to get $5 x = 2$ .
Divide by 5.
$x = 2 / 5$ .
Solve $3 x - 1 = - 3$ .
2 marks
Add 1 to get $3 x = - 2$ .
Divide by 3.
$x = - 2 / 3$ .
Solve $(x + 1) / 4 = 3 / 2$ .
2 marks
Multiply both sides by 4: $x + 1 = 6$ .
Subtract 1.
$x = 5$ .
Solve $0.5 x + 1.2 = 3.7$ .
2 marks
Subtract 1.2: $0.5 x = 2.5$ .
Divide by 0.5.
$x = 5$ .
Solve $x / 2 + x / 3 = 5$ .
3 marks
Multiply every term by 6: $3 x + 2 x = 30$ .
Then $5 x = 30$ ; check $3 + 2 = 5$ .
$x = 6$ .

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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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