Rearranging simple formulas | Year 8

01 / LEARN

Understand the idea.

Change the subject

The subject of a formula is the variable written alone on one side. In $C = 4 n + 7$ , making n the subject means undoing the operations on n in reverse order: first remove the added seven, then remove multiplication by four.

Apply each inverse operation to both sides to preserve equality. Dividing $C - 7$ by four means dividing the entire difference, so write $\frac{C - 7}{4}$ . The formula still describes the same relationship; only the variable that is isolated has changed.

Use and check the rearranged rule

From $A = l w$ , divide both sides by l to obtain $w = \frac{A}{l}$ when $l \neq 0$ . Division by zero is undefined, so any variable used as a divisor needs a restriction. A physical rectangle already has a positive length.

Substitute the measurements after rearranging and keep their units. Area divided by length gives length, as a width requires. Substituting the answer into the original formula checks the calculation; the equal operations justify the rearranged formula itself.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Undo addition then multiplication

Make $n$ the subject of $C = 4 n + 7$ .

Worked solution

1
Undo adding seven on both sides.
$C - 7 = 4 n$
Subtracting the same number from each side preserves the equality.
2
Undo multiplication by four.
$\frac{C - 7}{4} = n$
The whole left-hand difference is divided by four, which is nonzero.
3
Write the requested subject on the left.
$n = \frac{C - 7}{4}$
Reversing the sides of an equality does not change its meaning.

Answer

$n = \frac{C - 7}{4}$

EXAMPLE 02

Use a rearranged area formula

From $A = l w$ , find $w$ if $A = 42$ square centimetres and $l = 7$ cm.

Worked solution

1
Isolate width by dividing by length.
$w = \frac{A}{l}, l \neq 0$
The known length multiplies the unknown width in the original area formula.
2
Substitute the measurements.
$w = \frac{42 {cm}^{2}}{7 cm} = 6 cm$
Dividing square centimetres by centimetres leaves a length in centimetres.
3
Check the original area formula.
$7 cm \times 6 cm = 42 {cm}^{2}$
The calculated width and supplied length reproduce the given area.

Answer

$6$ cm

03 / YOUR TURN

Try it yourself.

Check your understanding

Make $x$ the subject of $y = 3 x - 2$ .

Show answer & reasoning

$x = \frac{y + 2}{3}$

Add two to both sides to obtain $y + 2 = 3 x$ . Then divide the complete sum by three, giving $x = \frac{y + 2}{3}$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Make $x$ the subject of $y = x + 8$ .
2 marks
Make $p$ the subject of $C = 5 p$ .
2 marks
Make $t$ the subject of $d = v t$ , where $v \neq 0$ .
2 marks
Make $n$ the subject of $T = 3 n + 6$ .
2 marks
Make $a$ the subject of $P = 2 a + 2 b$ .
2 marks
A repair costs $C = 6 h + 18$ dollars. Make $h$ the subject and find $h$ when $C = 54$ .
3 marks

Read the worked solutions

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

Make $x$ the subject of $y = x + 8$ .
2 marks
Subtract $8$ from both sides.
$x = y - 8$
Make $p$ the subject of $C = 5 p$ .
2 marks
Divide both sides by $5$ .
$p = \frac{C}{5}$
Make $t$ the subject of $d = v t$ , where $v \neq 0$ .
2 marks
Divide both sides by $v$ .
$t = \frac{d}{v}$
Make $n$ the subject of $T = 3 n + 6$ .
2 marks
Subtract $6$ , then divide by $3$ .
$n = \frac{T - 6}{3}$
Make $a$ the subject of $P = 2 a + 2 b$ .
2 marks
Subtract $2 b$ , then divide the full difference by $2$ .
$a = \frac{P - 2 b}{2}$
A repair costs $C = 6 h + 18$ dollars. Make $h$ the subject and find $h$ when $C = 54$ .
3 marks
Subtract $18$ and divide by $6$ .
$\frac{54 - 18}{6} = 6$ .
$h = \frac{C - 18}{6}$ ; $6$ hours.

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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.

Change the subject

Use and check the rearranged rule

See the method in action.

Undo addition then multiplication

Undo adding seven on both sides.

Undo multiplication by four.

Write the requested subject on the left.

Use a rearranged area formula

Isolate width by dividing by length.

Substitute the measurements.

Check the original area formula.

Try it yourself.

Check your understanding

Keep practising