Rearranging formulas | Year 10

01 / LEARN

The idea, step by step

Undo the outer operations

Rearranging a formula uses the same equality-preserving operations as solving an equation. Work with symbols rather than replacing them by numbers. Undo operations in a sensible order and show brackets clearly when an entire expression must be divided.

Collect a repeated subject

If the subject appears in several terms, collect those terms on one side and factor it out. Then divide by its coefficient. State any condition required by the division. Some parameter choices give a special case instead of a formula with a denominator.

A common trap

From $A = b h / 2$ , writing $h = A / (2 b)$ .

Instead: First multiply both sides by $2$ , then divide by $b$ : $h = 2 A / b$ , for $b \neq 0$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Acceleration as the subject

Make $a$ the subject of $v = u + a t$ .

Subtract $u$ : $v - u = a t$ .
Divide by $t$ , provided $t \neq 0$ .

a = \frac{v - u}{t}

t \neq 0

Example 2

The subject appears twice

Make $x$ the subject of $y = 3 x + a x$ .

Factor the right side: $y = x (3 + a)$ .
Divide by $a + 3$ when $a \neq - 3$ .
If $a = - 3$ , the original equation becomes $y = 0$ and needs separate interpretation.

x = \frac{y}{a + 3}

for

a \neq - 3

Check your understanding

Make $x$ the subject of $y = 5 x - 7$ .

Show answer

$x = (y + 7) / 5$

Add $7$ to both sides, then divide the entire result by $5$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Make $r$ the subject of $C = 2 π r$ .
2 marks
Make $h$ the subject of $A = b h / 2$ .
2 marks
Make $b$ the subject of $P = 2 a + 2 b$ .
2 marks
Make $x$ the subject of $y = m (x - 2) + c$ .
2 marks
Make $t$ the subject of $s = u t + v t$ .
2 marks
A trapezium has area $A = (a + b) h / 2$ . Make $b$ the subject, then find $b$ for $A = 45, a = 4, h = 6$ .
3 marks

Read the worked solutions

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

Make $r$ the subject of $C = 2 π r$ .
2 marks
Divide by the constant $2 π$ .
$r = C / (2 π)$
Make $h$ the subject of $A = b h / 2$ .
2 marks
Multiply by $2$ , then divide by $b$ .
$h = 2 A / b$ , $b \neq 0$
Make $b$ the subject of $P = 2 a + 2 b$ .
2 marks
Subtract $2 a$ , then divide by $2$ .
$b = (P - 2 a) / 2$
Make $x$ the subject of $y = m (x - 2) + c$ .
2 marks
Subtract $c$ , divide by $m$ , then add $2$ .
$x = (y - c) / m + 2$ , $m \neq 0$
Make $t$ the subject of $s = u t + v t$ .
2 marks
Factor $t$ : $s = t (u + v)$ .
$t = s / (u + v)$ when $u + v \neq 0$ .
A trapezium has area $A = (a + b) h / 2$ . Make $b$ the subject, then find $b$ for $A = 45, a = 4, h = 6$ .
3 marks
$2 A = (a + b) h$ , then $2 A / h = a + b$ with $h \neq 0$ .
Substitute: $90 / 6 - 4 = 11$ .
$b = 2 A / h - a$ ; $b = 11$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian Mathematics Curriculum

Year 10 algebra, including the approved fuller school-course treatment of rational expressions and formula restrictions; quadratic methods are in their own chapter.

Official curriculum source ↗

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