YEAR 8 · LINEAR EQUATIONS AND GRAPHS · 15 MIN

Multi-step equations

Undo several operations while keeping both sides of an equation equal.

In this lesson

Multi-step equations
Explain a method and check its result.

Before you start

One-step equations
Order of operations

01 / LEARN

Understand the idea.

What solving means

An equation says that two expressions have equal values. A solution is a value of the unknown that makes this true. Keep the balance by applying the same operation to both sides; never divide by zero.

First combine like terms, such as $5 x - 2 x = 3 x$ . Then undo the operations around the unknown in reverse order: remove an added constant before dividing by the coefficient multiplying the unknown.

Show equivalent statements

Each line should follow from a stated operation on both sides. For $3 x + 5 = 17$ , subtracting $5$ leaves $3 x = 12$ ; dividing by $3$ then isolates the unknown. These changes preserve the solution.

Check your answer in the original equation, where an earlier error cannot be hidden. Keep an exact fractional answer if one arises; changing it to a nearby whole number changes the value being checked.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Undo two operations

Solve $4 x + 7 = 31$ .

Worked solution

1
Subtract 7 from both sides.
$4 x = 31 - 7 = 24$
Removing the added constant leaves the unknown term by itself.
2
Divide both sides by 4.
$x = \frac{24}{4} = 6$
The coefficient $4$ multiplies the unknown, so division reverses that operation.
3
Substitute into the original equation.
$4 (6) + 7 = 24 + 7 = 31$
The original left side equals the right side, confirming $x = 6$ .

Answer

$x = 6$ .

EXAMPLE 02

Collect terms first

Solve $5 x + 8 - 2 x = 20$ .

Worked solution

1
Collect the like terms.
$(5 - 2) x + 8 = 20$
Both variable terms are multiples of $x$ , so subtract their coefficients.
2
Subtract 8 from both sides.
$3 x = 20 - 8 = 12$
The constant is removed without changing the balance.
3
Divide both sides by 3.
$x = \frac{12}{3} = 4$
Three equal copies of $x$ have total $12$ .
4
Check in the original expression.
$5 (4) + 8 - 2 (4) = 20 + 8 - 8 = 20$
Both original sides now have the same value.

Answer

$x = 4$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Solve $3 x - 5 = 16$ .

Show answer & reasoning

$x = 7$ .

Add $5$ to both sides to get $3 x = 21$ , then divide by $3$ to obtain $x = 7$ . Substitution gives $3 (7) - 5 = 16$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Solve $2 x + 5 = 17$ .
2 marks
Solve $5 x - 4 = 26$ .
2 marks
Solve $7 + 3 x = 1$ .
2 marks
Solve $6 x + 4 - 2 x = 24$ .
2 marks
Solve $\frac{x}{3} + 4 = 9$ .
2 marks
Bike hire costs AUD 6 plus AUD 4 per hour. The bill is AUD 26. Write and solve an equation for the hours.
3 marks

Read the worked solutions

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

Solve $2 x + 5 = 17$ .
2 marks
Subtract 5 to get $2 x = 12$ .
Divide by 2; check $12 + 5 = 17$ .
$x = 6$ .
Solve $5 x - 4 = 26$ .
2 marks
Add 4 to get $5 x = 30$ .
Divide by 5.
$x = 6$ .
Solve $7 + 3 x = 1$ .
2 marks
Subtract 7 to get $3 x = - 6$ .
Divide by 3 and check.
$x = - 2$ .
Solve $6 x + 4 - 2 x = 24$ .
2 marks
Collect terms: $4 x + 4 = 24$ .
Subtract 4 and divide by 4.
$x = 5$ .
Solve $\frac{x}{3} + 4 = 9$ .
2 marks
Subtract 4: $\frac{x}{3} = 5$ .
Multiply by 3.
$x = 15$ .
Bike hire costs AUD 6 plus AUD 4 per hour. The bill is AUD 26. Write and solve an equation for the hours.
3 marks
Remove the fixed charge: $4 h = 20$ .
Divide by 4 and check $6 + 4 (5) = 26$ .
$6 + 4 h = 26$ ; 5 hours.

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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 ↗

In this lesson

Understand the idea.

What solving means

Show equivalent statements

See the method in action.

Undo two operations

Subtract 7 from both sides.

Divide both sides by 4.

Substitute into the original equation.

Collect terms first

Collect the like terms.

Subtract 8 from both sides.

Divide both sides by 3.

Check in the original expression.

Try it yourself.

Check your understanding

Keep practising