Change of subject

Video lesson5 worked examples

Theory

Changing the subject involves algebraically rearranging a formula to isolate a different variable.

This applies to linear formulas, as well as those involving simple squares or square roots.

You are expected to apply this to real-life contexts using formulae from science, health, and finance (e.g., $E = \frac{1}{2}mv^2$ or calculating the radius of a sphere given its volume).

The Golden Rule: use inverse operations to peel away everything around the target variable, in reverse order (undo $+/-$ before $\times/\div$ , and undo those before powers/roots). To undo a square, take a root; to undo a root, square both sides. If the target appears in two terms, factorise it out first.

⚠️ Common Examiner Traps

Undoing in the wrong order: reverse the order of operations — deal with terms added or subtracted before multiplying or dividing.
Clearing a fraction: if the subject is trapped in a fraction, multiply both sides by the denominator first.
Roots and squares: to free a variable under a square root, square both sides; to free a squared variable, square-root both sides.
Subject in two terms: if the target appears twice (e.g. $ax + bx$ ), you must factorise it out — $x(a + b)$ — before you can divide.

Worked examples

Example 1

Linear Formula

Change the subject of $P = 2L + 2B$ to L.

Step 1: Subtract 2B from both sides: $P - 2B = 2L$ .

Step 2: Divide both sides by 2. Answer: $L = \dfrac{P - 2B}{2}$ .

Example 2

Freeing a Squared Variable

Change the subject of $A = \pi r^2$ to r.

Step 1: Divide by $\pi$ : $\dfrac{A}{\pi} = r^2$ .

Step 2: Undo the square by taking the square root of both sides. Answer: $r = \sqrt{\dfrac{A}{\pi}}$ .

Example 3

Subject in a Fraction

Change the subject of $T = \dfrac{2a + 5}{b}$ to a.

Step 1: The subject is trapped in a fraction, so multiply both sides by $b$ first: $Tb = 2a + 5$ .

Step 2: Subtract 5 from both sides: $Tb - 5 = 2a$ .

Step 3: Divide by 2. Answer: $a = \dfrac{Tb - 5}{2}$ .

Example 4

Freeing a Variable Under a Root

Change the subject of $y = \sqrt{3x + 1}$ to x.

Step 1: The $x$ is trapped under a square root, so undo it by squaring both sides: $y^2 = 3x + 1$ .

Step 2: Subtract 1 from both sides: $y^2 - 1 = 3x$ .

Step 3: Divide by 3. Answer: $x = \dfrac{y^2 - 1}{3}$ .

Example 5

🔗 Bringing it together

Change the subject of $ax + bx = c$ to x.

Step 1: The subject $x$ appears in two terms, so it cannot be isolated by dividing yet. Factorise it out:

x(a + b) = c

Step 2: Now $x$ is a single factor — divide both sides by the bracket:

Answer: $x = \dfrac{c}{a + b}$ .

Ready to practise?

18 Change of Subject questions with answers and video solutions