Changing the subject of a formula means rearranging a formula so that a different letter is on its own.
The key rule is:
Whatever you do to one side of the equation, you must do to the other side.
Think of it like solving an equation, except you are working with letters instead of just numbers.
Simple example
Change the subject of:
\[ y=x+5 \]
Make $x$ the subject.
Step 1: We want $x$ on its own.
Step 2: Subtract 5 from both sides.
\[ y-5=x+5-5 \]
Step 3: Simplify.
\[ \boxed{x=y-5} \]
Notice that the $+5$ became $-5$.
2. The four operations you need
| Operation | Opposite operation |
|---|---|
| Add 5 | Subtract 5 |
| Subtract 5 | Add 5 |
| Multiply by 5 | Divide by 5 |
| Divide by 5 | Multiply by 5 |
Always use the opposite operation to move something away from the letter you want to make the subject.
Example A: The letter is multiplied
Make $x$ the subject of:
\[ y=4x \]
Step 1: $x$ is multiplied by 4.
Step 2: Divide both sides by 4.
\[ \frac{y}{4}=\frac{4x}{4} \]
Answer:
\[ \boxed{x=\frac{y}{4}} \]
Example B: Two operations
Make $x$ the subject of:
\[ y=3x+7 \]
Step 1: Subtract 7 from both sides.
\[ y-7=3x \]
Step 2: Divide both sides by 3.
\[ \boxed{x=\frac{y-7}{3}} \]
Important: Undo the operations in reverse order. Subtract 7 first, then divide by 3.
Example C: The letter appears in a fraction
Make $x$ the subject of:
\[ y=\frac{x}{5}+2 \]
Step 1: Subtract 2.
\[ y-2=\frac{x}{5} \]
Step 2: Multiply both sides by 5.
\[ \boxed{x=5(y-2)} \]
Remember to multiply the whole bracket by 5.
A harder Grade 5 example
Make $x$ the subject of:
\[ y=2x+3 \]
This is the same method as Example B.
1. Subtract 3:
\[ y-3=2x \]
2. Divide by 2:
\[ \boxed{x=\frac{y-3}{2}} \]
When the letter appears on both sides
Make $x$ the subject of:
\[ y=3x+5 \]
This example is straightforward, but if the formula contains $x$ on both sides, collect the $x$ terms together first.
For example:
\[ y=3x+5 \]
\[ y-5=3x \]
\[ x=\frac{y-5}{3} \]
The key is to get the letter you want on its own.
Remember these exam tips
- Undo operations in reverse order.
- Use the opposite operation to move terms.
- If there are brackets, deal with the whole bracket carefully.
- Check your answer by substituting it back into the original formula.
Your main strategy: Get rid of addition and subtraction first, then multiplication and division.