Changing the Subject of a Formula

What you'll learn

What it means to make a letter the subject of a formula.
How to use inverse operations to rearrange formulae safely.
How to handle brackets, fractions, squares, cubes and square roots.
How to avoid common sign and denominator mistakes.

1. What does “make the subject” mean?

A formula is an equation that connects variables. A variable is a letter that represents a number.

Definition

Subject of a formula

The subject of a formula is the variable that is on its own on one side of the equals sign. For example, in $A = l w$ , the subject is $A$ .

When you are asked to “make $x$ the subject”, your final answer should look like:

\text{something}

something=x\text{something} = x

but it is usually clearest to put the subject on the left.

2. The balance method

The equals sign means both sides are equal. If you do something to one side, you must do the same thing to the other side.

Definition

Inverse operations

An inverse operation undoes another operation. Adding undoes subtracting, subtracting undoes adding, multiplying undoes dividing, and square rooting undoes squaring.

Key Idea

Main idea

Changing the subject is like solving an equation, but with letters instead of numbers. Undo the operations in reverse order until your chosen letter is alone.

Example

Making a letter the subject

Make $r$ the subject of $q = 6 r - 10$ .

A balance diagram shows that adding 10 to both sides and then dividing by 6 keeps the formula equal while isolating r.

Start with the formula and identify the term containing $r$ :
$q = 6 r - 10$
Undo the subtract 10 by adding 10 to both sides:
$q + 10 = 6 r$
Undo the multiply by 6 by dividing both sides by 6:
$\frac{q + 10}{6}$

Common Mistake

Changing sides too quickly

Do not just “move” a term and hope the sign changes correctly. Think: “What operation will undo this?” For example, subtracting 10 is undone by adding 10 to both sides.

3. Other letters are treated like numbers

A term is a part of an expression separated by plus or minus signs. For example, in $2 m + 5 p$ , the terms are $2 m$ and $5 p$ .

When rearranging, letters that are not the subject are treated like fixed values.

Example

A formula with two letter terms

Make $p$ the subject of $S = 2 m + 5 p$ .

The diagram highlights the p-term first, then shows subtracting 2m and dividing by 5 to leave p alone.

Identify the term containing $p$ :
$S = 2 m + 5 p$
Undo the add $2 m$ by subtracting $2 m$ from both sides:
$S - 2 m = 5 p$
Undo the multiply by 5 by dividing both sides by 5:
$\frac{S - 2m}{5}$

Tip

Keep the whole expression together

If you divide by 5, the whole of $S - 2 m$ is divided by 5. That is why the fraction line goes under both terms.

4. Brackets: undo the outside first

If the subject is inside brackets, it is often easiest to undo what is happening outside the brackets first.

Example

Subject inside brackets

Make $a$ the subject of $z = 4 (a + 7)$ .

The outside operation on the bracket is undone first, before removing the +7 inside the bracket.

Start with the formula:
$z = 4 (a + 7)$
Undo the multiply by 4 by dividing both sides by 4:
$z4=a+7\frac{z}{4} = a + 7$
Undo the add 7 by subtracting 7 from both sides:
$\frac{z}{4} - 7$

Tip

Brackets shortcut

If the whole bracket is being multiplied, divide first. Expanding the bracket also works, but it can create extra steps.

5. Fractions: clear the denominator

The denominator is the bottom part of a fraction. If a formula has a fraction, a good first move is often to multiply by the denominator.

Example

Subject in the numerator

Make $x$ the subject of $\frac{3x - 2}{5}$ .

Clearing the denominator affects the whole numerator, so 3x − 2 stays together before the remaining operations are undone.

Multiply both sides by 5 to remove the denominator:
$5 y = 3 x - 2$
Undo the subtract 2 by adding 2 to both sides:
$5 y + 2 = 3 x$
Undo the multiply by 3 by dividing both sides by 3:
$\frac{5y + 2}{3}$

Common Mistake

Only clearing part of the fraction

For $\frac{3x - 2}{5}$ , multiplying by 5 gives $5 y = 3 x - 2$ . Do not multiply only one term in the numerator.

Sometimes the subject starts in the denominator. First multiply by the denominator to bring it out of the fraction.

Example

Subject in the denominator

Make $s$ the subject of $\frac{r + 4}{s}$ .

Multiplying by s brings the subject out of the denominator before dividing by k isolates it.

Multiply both sides by $s$ :
$k s = r + 4$
Divide both sides by $k$ to get $s$ on its own:
$\frac{r + 4}{k}$

Common Mistake

Dividing by a letter

When you divide by a letter such as $k$ , the rearranged formula only works when that letter is not zero.

6. Powers: squares and cubes

A square means a value is multiplied by itself, such as $n^2$ . A square root undoes a square.

Example

Making the squared letter the subject

Make $n$ the subject of $T = n^2 + 6$ .

The diagram shows subtracting 6 first, then taking the square root and remembering both positive and negative roots.

Undo the add 6 by subtracting 6 from both sides:
$T - 6 = n^2$
Undo the square by square rooting both sides:
$\pm\sqrt{T - 6}$

Common Mistake

Forgetting the plus or minus

If $n^2 = 25$ , then $n$ could be 5 or -5. So when you square root an expression involving a squared variable, use $±\pm$ unless the question gives a reason to choose only the positive value.

A cube means a value is multiplied by itself three times, such as $x^3$ . A cube root undoes a cube.

Example

Making a cubed letter the subject

Make $x$ the subject of $y = x^3 + 4$ .

For a cubed subject, subtract the added term first and then apply the cube root.

Undo the add 4 by subtracting 4 from both sides:
$y - 4 = x^3$
Undo the cube by taking the cube root:
$\sqrt[3]{y - 4}$

7. Square roots around fractions

If the formula begins with a square root, square both sides first. This removes the square root.

Example

Subject under a square root

Make $h$ the subject of $\sqrt{\frac{h + 3}{4}}$ .

The outer square root is removed first, then the denominator is cleared and the +3 is undone.

Square both sides to remove the square root:
$r2=h+34r^2 = \frac{h + 3}{4}$
Multiply both sides by 4 to remove the denominator:
$4r^2 = h + 3$
Subtract 3 from both sides:
$h = 4r^2 - 3$

Tip

Outer layer first

Look at what is happening to the subject from the outside in. If everything is inside a square root, square first. If everything is over a denominator, multiply by the denominator first.

Exam technique

In the exam

Circle the letter you are making the subject, so you know your target.
Undo operations one at a time, writing a clear new line after each move.
Check your final line: the subject letter should appear once, on its own, with no hidden brackets, powers or denominators attached to it.

Self review

Check yourself

If $y = 7 x - 3$ , what operation would you undo first to make $x$ the subject?
Why does $\frac{2a + 5}{3}$ become $3 w = 2 a + 5$ after clearing the fraction?
When rearranging $m = p^2 - 1$ , why might the final answer need a plus or minus sign?

Changing the Subject of a Formula

What you'll learn

What it means to make a letter the subject of a formula.
How to use inverse operations to rearrange formulae safely.
How to handle brackets, fractions, squares, cubes and square roots.
How to avoid common sign and denominator mistakes.

1. What does “make the subject” mean?

A formula is an equation that connects variables. A variable is a letter that represents a number.

Definition

Subject of a formula

The subject of a formula is the variable that is on its own on one side of the equals sign. For example, in $A = l w$ , the subject is $A$ .

When you are asked to “make $x$ the subject”, your final answer should look like:

\text{something}

something=x\text{something} = x

but it is usually clearest to put the subject on the left.

2. The balance method

The equals sign means both sides are equal. If you do something to one side, you must do the same thing to the other side.

Definition

Inverse operations

An inverse operation undoes another operation. Adding undoes subtracting, subtracting undoes adding, multiplying undoes dividing, and square rooting undoes squaring.

Key Idea

Main idea

Changing the subject is like solving an equation, but with letters instead of numbers. Undo the operations in reverse order until your chosen letter is alone.

Example

Making a letter the subject

Make $r$ the subject of $q = 6 r - 10$ .

A balance diagram shows that adding 10 to both sides and then dividing by 6 keeps the formula equal while isolating r.

Start with the formula and identify the term containing $r$ :
$q = 6 r - 10$
Undo the subtract 10 by adding 10 to both sides:
$q + 10 = 6 r$
Undo the multiply by 6 by dividing both sides by 6:
$\frac{q + 10}{6}$

Common Mistake

Changing sides too quickly

Do not just “move” a term and hope the sign changes correctly. Think: “What operation will undo this?” For example, subtracting 10 is undone by adding 10 to both sides.

3. Other letters are treated like numbers

A term is a part of an expression separated by plus or minus signs. For example, in $2 m + 5 p$ , the terms are $2 m$ and $5 p$ .

When rearranging, letters that are not the subject are treated like fixed values.

Example

A formula with two letter terms

Make $p$ the subject of $S = 2 m + 5 p$ .

The diagram highlights the p-term first, then shows subtracting 2m and dividing by 5 to leave p alone.

Identify the term containing $p$ :
$S = 2 m + 5 p$
Undo the add $2 m$ by subtracting $2 m$ from both sides:
$S - 2 m = 5 p$
Undo the multiply by 5 by dividing both sides by 5:
$\frac{S - 2m}{5}$

Tip

Keep the whole expression together

If you divide by 5, the whole of $S - 2 m$ is divided by 5. That is why the fraction line goes under both terms.

4. Brackets: undo the outside first

If the subject is inside brackets, it is often easiest to undo what is happening outside the brackets first.

Example

Subject inside brackets

Make $a$ the subject of $z = 4 (a + 7)$ .

The outside operation on the bracket is undone first, before removing the +7 inside the bracket.

Start with the formula:
$z = 4 (a + 7)$
Undo the multiply by 4 by dividing both sides by 4:
$z4=a+7\frac{z}{4} = a + 7$
Undo the add 7 by subtracting 7 from both sides:
$\frac{z}{4} - 7$

Tip

Brackets shortcut

If the whole bracket is being multiplied, divide first. Expanding the bracket also works, but it can create extra steps.

5. Fractions: clear the denominator

The denominator is the bottom part of a fraction. If a formula has a fraction, a good first move is often to multiply by the denominator.

Example

Subject in the numerator

Make $x$ the subject of $\frac{3x - 2}{5}$ .

Clearing the denominator affects the whole numerator, so 3x − 2 stays together before the remaining operations are undone.

Multiply both sides by 5 to remove the denominator:
$5 y = 3 x - 2$
Undo the subtract 2 by adding 2 to both sides:
$5 y + 2 = 3 x$
Undo the multiply by 3 by dividing both sides by 3:
$\frac{5y + 2}{3}$

Common Mistake

Only clearing part of the fraction

For $\frac{3x - 2}{5}$ , multiplying by 5 gives $5 y = 3 x - 2$ . Do not multiply only one term in the numerator.

Sometimes the subject starts in the denominator. First multiply by the denominator to bring it out of the fraction.

Example

Subject in the denominator

Make $s$ the subject of $\frac{r + 4}{s}$ .

Multiplying by s brings the subject out of the denominator before dividing by k isolates it.

Multiply both sides by $s$ :
$k s = r + 4$
Divide both sides by $k$ to get $s$ on its own:
$\frac{r + 4}{k}$

Common Mistake

Dividing by a letter

When you divide by a letter such as $k$ , the rearranged formula only works when that letter is not zero.

6. Powers: squares and cubes

A square means a value is multiplied by itself, such as $n^2$ . A square root undoes a square.

Example

Making the squared letter the subject

Make $n$ the subject of $T = n^2 + 6$ .

The diagram shows subtracting 6 first, then taking the square root and remembering both positive and negative roots.

Undo the add 6 by subtracting 6 from both sides:
$T - 6 = n^2$
Undo the square by square rooting both sides:
$\pm\sqrt{T - 6}$

Common Mistake

Forgetting the plus or minus

If $n^2 = 25$ , then $n$ could be 5 or -5. So when you square root an expression involving a squared variable, use $±\pm$ unless the question gives a reason to choose only the positive value.

A cube means a value is multiplied by itself three times, such as $x^3$ . A cube root undoes a cube.

Example

Making a cubed letter the subject

Make $x$ the subject of $y = x^3 + 4$ .

For a cubed subject, subtract the added term first and then apply the cube root.

Undo the add 4 by subtracting 4 from both sides:
$y - 4 = x^3$
Undo the cube by taking the cube root:
$\sqrt[3]{y - 4}$

7. Square roots around fractions

If the formula begins with a square root, square both sides first. This removes the square root.

Example

Subject under a square root

Make $h$ the subject of $\sqrt{\frac{h + 3}{4}}$ .

The outer square root is removed first, then the denominator is cleared and the +3 is undone.

Square both sides to remove the square root:
$r2=h+34r^2 = \frac{h + 3}{4}$
Multiply both sides by 4 to remove the denominator:
$4r^2 = h + 3$
Subtract 3 from both sides:
$h = 4r^2 - 3$

Tip

Outer layer first

Look at what is happening to the subject from the outside in. If everything is inside a square root, square first. If everything is over a denominator, multiply by the denominator first.

Exam technique

In the exam

Circle the letter you are making the subject, so you know your target.
Undo operations one at a time, writing a clear new line after each move.
Check your final line: the subject letter should appear once, on its own, with no hidden brackets, powers or denominators attached to it.

Self review

Check yourself

If $y = 7 x - 3$ , what operation would you undo first to make $x$ the subject?
Why does $\frac{2a + 5}{3}$ become $3 w = 2 a + 5$ after clearing the fraction?
When rearranging $m = p^2 - 1$ , why might the final answer need a plus or minus sign?

Changing the Subject of a Formula

What you'll learn

1. What does “make the subject” mean?

2. The balance method

3. Other letters are treated like numbers

4. Brackets: undo the outside first

5. Fractions: clear the denominator

6. Powers: squares and cubes

7. Square roots around fractions

Recap questions

You've reached the end

Changing the Subject of a Formula

What you'll learn

1. What does “make the subject” mean?

2. The balance method

3. Other letters are treated like numbers

4. Brackets: undo the outside first

5. Fractions: clear the denominator

6. Powers: squares and cubes

7. Square roots around fractions

Recap questions

You've reached the end

Changing the Subject of a Formula

What you'll learn

1. What does “make the subject” mean?

2. The balance method

3. Other letters are treated like numbers

4. Brackets: undo the outside first

5. Fractions: clear the denominator

6. Powers: squares and cubes

7. Square roots around fractions

Recap questions

You've reached the end

Changing the Subject of a Formula Revision Guide

Changing the Subject of a Formula

What you'll learn

1. What does “make the subject” mean?

2. The balance method

3. Other letters are treated like numbers

4. Brackets: undo the outside first

5. Fractions: clear the denominator

6. Powers: squares and cubes

7. Square roots around fractions

Recap questions

You've reached the end

Changing the Subject of a Formula Revision Guide