YEAR 10 · ALGEBRA · 15 MIN

Rearranging formulas

Make a chosen variable the subject by undoing operations, including formulas where the variable appears more than once.

In this lesson

Isolate a chosen variable symbolically
Identify when division creates a condition

Before you start

Inverse operations
Common factors

01 / LEARN

Understand the idea.

Undo the outer operations

To make a variable the subject, isolate it using operations that preserve equality. Work backwards through the operations acting on it: undo an outside addition before an inner multiplication. A fraction bar must cover the whole expression being divided.

Division by a symbolic expression requires a condition that it is nonzero. Record that condition beside the rearranged formula; a parameter that makes the divisor zero must be considered separately in the original equation.

Collect a repeated subject

When the required subject appears in several terms, collect those terms on one side, then factor the subject out. Its remaining bracket is the coefficient to divide by. Avoid dividing individual terms before collecting them, which can leave the subject on both sides.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Acceleration as the subject

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

Worked solution

1
Remove the added initial value.
$v - u = a t$
Subtract $u$ from both sides of $v = u + a t$ .
2
Divide by the coefficient of acceleration.
$a = \frac{v - u}{t}, t \neq 0$
Division by $t$ isolates $a$ only when $t$ is nonzero.
3
Interpret the zero-time exception.
$t = 0 ⟹ v = u$
At zero time the original equation contains no $a$ , so it cannot determine a unique acceleration.

Answer

$a = \frac{v - u}{t}$ , $t \neq 0$

EXAMPLE 02

The subject appears twice

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

Worked solution

1
Factor the repeated subject.
$y = 3 x + a x = x (3 + a)$
Both terms on the right contain the factor $x$ .
2
Divide by its combined coefficient.
$x = \frac{y}{a + 3}, a \neq - 3$
The factor $a + 3$ may be divided out only when it is nonzero.
3
Describe the exceptional parameter value.
$a = - 3 ⟹ y = 0$
If $a = - 3$ , every $x$ works when $y = 0$ , while no $x$ works when $y \neq 0$ .

Answer

$x = \frac{y}{a + 3}$ for $a \neq - 3$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$x = (y + 7) / 5$

Add $7$ to both sides to obtain $y + 7 = 5 x$ . Divide the whole left side by $5$ : $x = \frac{y + 7}{5}$ . The numerator needs brackets or a fraction bar covering both terms.

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

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

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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 ↗

In this lesson

Understand the idea.

Undo the outer operations

Collect a repeated subject

See the method in action.

Acceleration as the subject

Remove the added initial value.

Divide by the coefficient of acceleration.

Interpret the zero-time exception.

The subject appears twice

Factor the repeated subject.

Divide by its combined coefficient.

Describe the exceptional parameter value.

Try it yourself.

Check your understanding

Keep practising