Changing the Subject of a Formula

The Goal

Changing the subject means rearranging a formula so that a specific letter sits alone on one side of the equals sign.

If we want to "make $x$ the subject", our final answer must look like:

$$ x = \dots $$

1. The Golden Rule: Reverse Operations

Think of the subject (let's say $x$) as a gift that has been wrapped up in layers. To get to the $x$, you must unwrap the layers one by one, starting from the outside.

You do this by performing the Opposite Operation:

If you see Addition (+), you Subtract.
If you see Subtraction (-), you Add.
If you see Multiplication ($\times$), you Divide.
If you see Division ($\div$), you Multiply.
If you see Squaring ($x^2$), you Square Root.

2. Reverse SAMDIB

When solving, we usually follow BIDMAS/BODMAS. When rearranging, we usually go in reverse:

Undo Addition/Subtraction first.
Undo Multiplication/Division next.
Undo Indices (Powers/Roots) last.

Example: Make $x$ the subject of $y = 2x + 5$ Step 1: Undo the +5 (Subtract 5 from both sides) $$ y - 5 = 2x $$ Step 2: Undo the $\times 2$ (Divide both sides by 2) $$ \frac{y - 5}{2} = x $$ Answer: $ x = \frac{y - 5}{2} $

3. The "Factorising" Trap

Sometimes $x$ appears twice in the equation. You cannot just move one $x$ and leave the other. You must collect them on the same side and factorise.

Example: Make $x$ the subject of $ax - y = bx$ Step 1: Get all $x$'s on one side. $$ ax - bx = y $$ Step 2: Factorise out the $x$. $$ x(a - b) = y $$ Step 3: Divide by the bracket. $$ x = \frac{y}{a - b} $$

Practice Questions & Solutions

Try to make $x$ the subject in each question.

Q1: $ y = x - 4 $

Opposite of minus 4 is plus 4.
Add 4 to both sides.
$ x = y + 4 $

Q2: $ y = 5x $

This means $5 \times x$.
Opposite of multiply is divide.
Divide both sides by 5.
$ x = \frac{y}{5} $

Q3: $ y = mx + c $

1. Move the $+c$ first (Subtraction).
$ y - c = mx $

2. Move the $m$ (Division).
$ x = \frac{y - c}{m} $

Q4: $ y = \frac{x}{a} - b $

1. Move the $-b$ (Addition).
$ y + b = \frac{x}{a} $

2. Move the $\div a$ (Multiplication).
Be careful: Multiply the entire left side by $a$.
$ x = a(y + b) $

Q5: $ y = x^2 + 7 $

1. Move the $+7$ (Subtraction).
$ y - 7 = x^2 $

2. Undo the square (Square Root).
$ x = \sqrt{y - 7} $

Q6: $ y = \sqrt{x + 3} $

1. Undo the square root (Square both sides).
$ y^2 = x + 3 $

2. Move the $+3$ (Subtraction).
$ x = y^2 - 3 $

Q7: $ y = \frac{5}{x} $

Trick Question: $x$ is on the bottom!
1. Multiply by $x$ to get it off the bottom.
$ xy = 5 $

2. Divide by $y$ to leave $x$ alone.
$ x = \frac{5}{y} $

Q8: $ P = 2(x + w) $

1. Divide by 2 (easier than expanding first).
$ \frac{P}{2} = x + w $

2. Subtract $w$.
$ x = \frac{P}{2} - w $

Q9: $ ax + b = cx + d $

$x$ is on both sides.
1. Subtract $cx$ to get $x$'s on the left.
$ ax - cx + b = d $

2. Subtract $b$ to get non-$x$'s on the right.
$ ax - cx = d - b $

3. Factorise the left side.
$ x(a - c) = d - b $

4. Divide by the bracket.
$ x = \frac{d - b}{a - c} $

Q10: $ y = \frac{x + a}{x - b} $

Hardest Type!
1. Multiply by the denominator $(x - b)$.
$ y(x - b) = x + a $

2. Expand the bracket.
$ xy - by = x + a $

3. Collect $x$ terms on the left, everything else on the right.
$ xy - x = a + by $

4. Factorise $x$.
$ x(y - 1) = a + by $

5. Divide by the bracket.
$ x = \frac{a + by}{y - 1} $