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 deal with brackets, fractions, squares, cubes and square roots.
How to avoid the most common sign and order mistakes in exam questions.

Formulae, variables and subjects

A formula is an equation that connects two or more quantities. A variable is a letter that stands for a number, such as $x$ , $t$ , $a$ or $b$ .

Definition

Subject of a formula

The subject of a formula is the variable written on its own, usually on the left-hand side.

For example, in $y = 3 x + 4$ , the subject is $y$ .

When a question says “make $x$ the subject”, your final answer should look like:

\text{something}

The goal is to isolate the required letter.

Example

Making a letter the subject in a simple linear formula

Make $r$ the subject of $w = 7 r - 9$ .

Identify the subject you want: $r$ . At the moment, $r$ has been multiplied by 7 and then 9 has been subtracted.
Undo subtracting 9 by adding 9 to both sides.
$w + 9 = 7 r$
Undo multiplying by 7 by dividing both sides by 7.
$w+97=r\frac{w + 9}{7} = r$
Write the final answer with $r$ on the left.
$\frac{w + 9}{7}$

Key Idea

Keep the equation balanced

Whatever you do to one side of an equation, you must do to the other side. This keeps the equation true.

Inverse operations: undoing the formula

An inverse operation is an operation that undoes another operation.

Definition

Inverse operations

Addition and subtraction are inverse operations.
Multiplication and division are inverse operations.
Squaring and square rooting are inverse operations.
Cubing and cube rooting are inverse operations.

When rearranging, think of the formula as a machine. To get back to the original variable, reverse the operations in the opposite order.

Flow diagram showing how to reverse operations when making a variable the subject

Example

Undoing operations in reverse order

Make $p$ the subject of $z = 4 p + 13$ .

Focus on what has happened to $p$ : it was multiplied by 4, then 13 was added.
Reverse the last operation first. Subtract 13 from both sides.
$z - 13 = 4 p$
Now reverse multiplying by 4. Divide both sides by 4.
$z−134=p\frac{z - 13}{4} = p$
Write the answer neatly.
$\frac{z - 13}{4}$

Common Mistake

Undoing in the wrong order

If $z = 4 p + 13$ , do not divide by 4 first. The whole right-hand side is not $4 (p + 13)$ ; only $p$ is multiplied by 4.

Rearranging when there is more than one term

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

If the variable you want is part of one term, move the other terms away first.

Example

Making one variable the subject when another variable is present

Make $a$ the subject of $Q = 6 a + 6 b$ .

The term containing $a$ is $6 a$ . Move the other term, $6 b$ , away first.
Subtract $6 b$ from both sides.
$Q - 6 b = 6 a$
Divide both sides by 6.
$Q−6b6=a\frac{Q - 6b}{6} = a$
Write the final answer.
$\frac{Q - 6b}{6}$

Tip

Factor if it makes the answer cleaner

In $Q = 6 a + 6 b$ , you could also write $Q = 6 (a + b)$ . Then divide by 6 first and subtract $b$ . Both methods give $\frac{Q}{6} - b$ .

Brackets: divide first, then remove the bracket

If the variable is inside brackets, first undo anything happening to the whole bracket.

Example

Variable inside brackets

Make $h$ the subject of $x = 5 (h - 2)$ .

The whole bracket $h - 2$ is multiplied by 5, so undo that first.
$x5=h−2\frac{x}{5} = h - 2$
Now undo subtracting 2 by adding 2 to both sides.
$x5+2=h\frac{x}{5} + 2 = h$
Write the final answer with $h$ on the left.
$\frac{x}{5} + 2$

Common Mistake

Only dividing one term

From $x = 5 (h - 2)$ , do not write $x = 5 h - 2$ . The 5 multiplies everything inside the bracket, so $5 (h - 2) = 5 h - 10$ .

Fractions: clear the denominator

The denominator is the bottom number or expression in a fraction. To remove a denominator, multiply both sides by it.

Example

Variable in the numerator of a fraction

Make $x$ the subject of $\frac{3x - 1}{5}$ .

The expression $3 x - 1$ is divided by 5, so multiply both sides by 5.
$5 y = 3 x - 1$
Add 1 to both sides.
$5 y + 1 = 3 x$
Divide both sides by 3.
$5y+13=x\frac{5y + 1}{3} = x$
Write the final answer.
$\frac{5y + 1}{3}$

Example

Variable in the denominator

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

Multiply both sides by $r$ to remove the denominator.
$k r = m + 4$
Divide both sides by $k$ .
$\frac{m + 4}{k}$

Common Mistake

Dividing by zero

A denominator cannot be zero. For example, in $\frac{m + 4}{r}$ , the formula only makes sense when $\neq 0$ .

Fractions with coefficients

A coefficient is the number multiplying a variable. In $23x\frac{2}{3}x$ , the coefficient of $x$ is $23\frac{2}{3}$ .

To undo multiplying by a fraction, multiply by its reciprocal. The reciprocal of $23\frac{2}{3}$ is $32\frac{3}{2}$ .

Example

Rearranging when the coefficient is a fraction

Make $x$ the subject of $\frac{3}{4}x - 8$ .

Add 8 to both sides to isolate the term containing $x$ .
$\frac{3}{4}x$
Multiply both sides by the reciprocal of $34\frac{3}{4}$ , which is $43\frac{4}{3}$ .
$43(y+8)=x\frac{4}{3}(y + 8) = x$
Write the answer with $x$ on the left.
$\frac{4(y + 8)}{3}$

Tip

Keep brackets when multiplying a whole expression

If you multiply $y + 8$ by $43\frac{4}{3}$ , use brackets: $43(y+8)\frac{4}{3}(y + 8)$ . This shows the whole expression is being multiplied.

Squares, cubes and roots

A square root undoes squaring. A cube root undoes cubing.

Example

Making a variable the subject when it is squared

Make $n$ the subject of $s = n^2 + 10$ .

Subtract 10 from both sides.
$s - 10 = n^2$
Square root both sides. Because both positive and negative numbers can square to the same value, include $±\pm$ unless the context says the variable is positive.
$\pm\sqrt{s - 10}$

Example

Making a variable the subject when it is cubed

Make $x$ the subject of $y = x^3 - 6$ .

Add 6 to both sides.
$y + 6 = x^3$
Cube root both sides.
$\sqrt[3]{y + 6}$

Common Mistake

Squares and square roots

If $n^2 = 25$ , then $n = 5$ or $n = - 5$ . That is why $±\pm$ is needed unless the question gives a reason to choose only the positive value.

Square roots with fractions inside

If a formula has a square root, undo the square root first by squaring both sides.

Example

Square root with a fraction inside

Make $q$ the subject of $\sqrt{\frac{q + 4}{7}}$ .

Square both sides to remove the square root.
$g2=q+47g^2 = \frac{q + 4}{7}$
Multiply both sides by 7.
$7g^2 = q + 4$
Subtract 4 from both sides.
$7g^2 - 4 = q$
Write the final answer.
$q = 7g^2 - 4$

Example

Square root with the required variable inside

Make $h$ the subject of $\sqrt{\frac{5h}{3}}$ .

Square both sides.
$D2=5h3D^2 = \frac{5h}{3}$
Multiply both sides by 3.
$3D^2 = 5h$
Divide both sides by 5.
$\frac{3D^2}{5}$

Formulae that look more like science

Some rearranging questions use formulae from mechanics or physics. The algebra is exactly the same.

Example

Making the acceleration-style variable the subject

Make $a$ the subject of $V = U + a T$ .

Subtract $U$ from both sides.
$V - U = a T$
Divide both sides by $T$ .
$V−UT=a\frac{V - U}{T} = a$
Write the final answer.
$\frac{V - U}{T}$

Example

Making a variable the subject in a squared formula

Make $k$ the subject of $A^2 = B^2 + 6kL$ .

Subtract $B^2$ from both sides.
$A^2 - B^2 = 6kL$
Divide both sides by $6 L$ .
$\frac{A^2 - B^2}{6L}$

Exam technique

In the exam

Circle the variable you are making the subject, then work out what operations are attached to it.
Undo operations in reverse order, writing one clear algebra line for each move.
Keep brackets around expressions like $y + 8$ when the whole expression is being multiplied or divided.
For squared variables, consider whether you need $±\pm$ in your final answer.

Self review

Check yourself

If $A = 4 b - 7$ , what would you do first to make $b$ the subject?
Why does $x = 3 (y + 5)$ not mean $x = 3 y + 5$ ?
When you square root both sides of $p^2 = q - 1$ , why might there be two possible answers?

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 deal with brackets, fractions, squares, cubes and square roots.
How to avoid the most common sign and order mistakes in exam questions.

Formulae, variables and subjects

A formula is an equation that connects two or more quantities. A variable is a letter that stands for a number, such as $x$ , $t$ , $a$ or $b$ .

Definition

Subject of a formula

The subject of a formula is the variable written on its own, usually on the left-hand side.

For example, in $y = 3 x + 4$ , the subject is $y$ .

When a question says “make $x$ the subject”, your final answer should look like:

\text{something}

The goal is to isolate the required letter.

Example

Making a letter the subject in a simple linear formula

Make $r$ the subject of $w = 7 r - 9$ .

Identify the subject you want: $r$ . At the moment, $r$ has been multiplied by 7 and then 9 has been subtracted.
Undo subtracting 9 by adding 9 to both sides.
$w + 9 = 7 r$
Undo multiplying by 7 by dividing both sides by 7.
$w+97=r\frac{w + 9}{7} = r$
Write the final answer with $r$ on the left.
$\frac{w + 9}{7}$

Key Idea

Keep the equation balanced

Whatever you do to one side of an equation, you must do to the other side. This keeps the equation true.

Inverse operations: undoing the formula

An inverse operation is an operation that undoes another operation.

Definition

Inverse operations

Addition and subtraction are inverse operations.
Multiplication and division are inverse operations.
Squaring and square rooting are inverse operations.
Cubing and cube rooting are inverse operations.

When rearranging, think of the formula as a machine. To get back to the original variable, reverse the operations in the opposite order.

Flow diagram showing how to reverse operations when making a variable the subject

Example

Undoing operations in reverse order

Make $p$ the subject of $z = 4 p + 13$ .

Focus on what has happened to $p$ : it was multiplied by 4, then 13 was added.
Reverse the last operation first. Subtract 13 from both sides.
$z - 13 = 4 p$
Now reverse multiplying by 4. Divide both sides by 4.
$z−134=p\frac{z - 13}{4} = p$
Write the answer neatly.
$\frac{z - 13}{4}$

Common Mistake

Undoing in the wrong order

If $z = 4 p + 13$ , do not divide by 4 first. The whole right-hand side is not $4 (p + 13)$ ; only $p$ is multiplied by 4.

Rearranging when there is more than one term

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

If the variable you want is part of one term, move the other terms away first.

Example

Making one variable the subject when another variable is present

Make $a$ the subject of $Q = 6 a + 6 b$ .

The term containing $a$ is $6 a$ . Move the other term, $6 b$ , away first.
Subtract $6 b$ from both sides.
$Q - 6 b = 6 a$
Divide both sides by 6.
$Q−6b6=a\frac{Q - 6b}{6} = a$
Write the final answer.
$\frac{Q - 6b}{6}$

Tip

Factor if it makes the answer cleaner

In $Q = 6 a + 6 b$ , you could also write $Q = 6 (a + b)$ . Then divide by 6 first and subtract $b$ . Both methods give $\frac{Q}{6} - b$ .

Brackets: divide first, then remove the bracket

If the variable is inside brackets, first undo anything happening to the whole bracket.

Example

Variable inside brackets

Make $h$ the subject of $x = 5 (h - 2)$ .

The whole bracket $h - 2$ is multiplied by 5, so undo that first.
$x5=h−2\frac{x}{5} = h - 2$
Now undo subtracting 2 by adding 2 to both sides.
$x5+2=h\frac{x}{5} + 2 = h$
Write the final answer with $h$ on the left.
$\frac{x}{5} + 2$

Common Mistake

Only dividing one term

From $x = 5 (h - 2)$ , do not write $x = 5 h - 2$ . The 5 multiplies everything inside the bracket, so $5 (h - 2) = 5 h - 10$ .

Fractions: clear the denominator

The denominator is the bottom number or expression in a fraction. To remove a denominator, multiply both sides by it.

Example

Variable in the numerator of a fraction

Make $x$ the subject of $\frac{3x - 1}{5}$ .

The expression $3 x - 1$ is divided by 5, so multiply both sides by 5.
$5 y = 3 x - 1$
Add 1 to both sides.
$5 y + 1 = 3 x$
Divide both sides by 3.
$5y+13=x\frac{5y + 1}{3} = x$
Write the final answer.
$\frac{5y + 1}{3}$

Example

Variable in the denominator

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

Multiply both sides by $r$ to remove the denominator.
$k r = m + 4$
Divide both sides by $k$ .
$\frac{m + 4}{k}$

Common Mistake

Dividing by zero

A denominator cannot be zero. For example, in $\frac{m + 4}{r}$ , the formula only makes sense when $\neq 0$ .

Fractions with coefficients

A coefficient is the number multiplying a variable. In $23x\frac{2}{3}x$ , the coefficient of $x$ is $23\frac{2}{3}$ .

To undo multiplying by a fraction, multiply by its reciprocal. The reciprocal of $23\frac{2}{3}$ is $32\frac{3}{2}$ .

Example

Rearranging when the coefficient is a fraction

Make $x$ the subject of $\frac{3}{4}x - 8$ .

Add 8 to both sides to isolate the term containing $x$ .
$\frac{3}{4}x$
Multiply both sides by the reciprocal of $34\frac{3}{4}$ , which is $43\frac{4}{3}$ .
$43(y+8)=x\frac{4}{3}(y + 8) = x$
Write the answer with $x$ on the left.
$\frac{4(y + 8)}{3}$

Tip

Keep brackets when multiplying a whole expression

If you multiply $y + 8$ by $43\frac{4}{3}$ , use brackets: $43(y+8)\frac{4}{3}(y + 8)$ . This shows the whole expression is being multiplied.

Squares, cubes and roots

A square root undoes squaring. A cube root undoes cubing.

Example

Making a variable the subject when it is squared

Make $n$ the subject of $s = n^2 + 10$ .

Subtract 10 from both sides.
$s - 10 = n^2$
Square root both sides. Because both positive and negative numbers can square to the same value, include $±\pm$ unless the context says the variable is positive.
$\pm\sqrt{s - 10}$

Example

Making a variable the subject when it is cubed

Make $x$ the subject of $y = x^3 - 6$ .

Add 6 to both sides.
$y + 6 = x^3$
Cube root both sides.
$\sqrt[3]{y + 6}$

Common Mistake

Squares and square roots

If $n^2 = 25$ , then $n = 5$ or $n = - 5$ . That is why $±\pm$ is needed unless the question gives a reason to choose only the positive value.

Square roots with fractions inside

If a formula has a square root, undo the square root first by squaring both sides.

Example

Square root with a fraction inside

Make $q$ the subject of $\sqrt{\frac{q + 4}{7}}$ .

Square both sides to remove the square root.
$g2=q+47g^2 = \frac{q + 4}{7}$
Multiply both sides by 7.
$7g^2 = q + 4$
Subtract 4 from both sides.
$7g^2 - 4 = q$
Write the final answer.
$q = 7g^2 - 4$

Example

Square root with the required variable inside

Make $h$ the subject of $\sqrt{\frac{5h}{3}}$ .

Square both sides.
$D2=5h3D^2 = \frac{5h}{3}$
Multiply both sides by 3.
$3D^2 = 5h$
Divide both sides by 5.
$\frac{3D^2}{5}$

Formulae that look more like science

Some rearranging questions use formulae from mechanics or physics. The algebra is exactly the same.

Example

Making the acceleration-style variable the subject

Make $a$ the subject of $V = U + a T$ .

Subtract $U$ from both sides.
$V - U = a T$
Divide both sides by $T$ .
$V−UT=a\frac{V - U}{T} = a$
Write the final answer.
$\frac{V - U}{T}$

Example

Making a variable the subject in a squared formula

Make $k$ the subject of $A^2 = B^2 + 6kL$ .

Subtract $B^2$ from both sides.
$A^2 - B^2 = 6kL$
Divide both sides by $6 L$ .
$\frac{A^2 - B^2}{6L}$

Exam technique

In the exam

Circle the variable you are making the subject, then work out what operations are attached to it.
Undo operations in reverse order, writing one clear algebra line for each move.
Keep brackets around expressions like $y + 8$ when the whole expression is being multiplied or divided.
For squared variables, consider whether you need $±\pm$ in your final answer.

Self review

Check yourself

If $A = 4 b - 7$ , what would you do first to make $b$ the subject?
Why does $x = 3 (y + 5)$ not mean $x = 3 y + 5$ ?
When you square root both sides of $p^2 = q - 1$ , why might there be two possible answers?

Changing the Subject of a Formula

What you'll learn

Formulae, variables and subjects

Inverse operations: undoing the formula

Rearranging when there is more than one term

Brackets: divide first, then remove the bracket

Fractions: clear the denominator

Fractions with coefficients

Squares, cubes and roots

Square roots with fractions inside

Formulae that look more like science

Recap questions

You've reached the end

Changing the Subject of a Formula

What you'll learn

Formulae, variables and subjects

Inverse operations: undoing the formula

Rearranging when there is more than one term

Brackets: divide first, then remove the bracket

Fractions: clear the denominator

Fractions with coefficients

Squares, cubes and roots

Square roots with fractions inside

Formulae that look more like science

Recap questions

You've reached the end

Changing the Subject of a Formula

What you'll learn

Formulae, variables and subjects

Inverse operations: undoing the formula

Rearranging when there is more than one term

Brackets: divide first, then remove the bracket

Fractions: clear the denominator

Fractions with coefficients

Squares, cubes and roots

Square roots with fractions inside

Formulae that look more like science

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

Formulae, variables and subjects

Inverse operations: undoing the formula

Rearranging when there is more than one term

Brackets: divide first, then remove the bracket

Fractions: clear the denominator

Fractions with coefficients

Squares, cubes and roots

Square roots with fractions inside

Formulae that look more like science

Recap questions

You've reached the end

Changing the Subject of a Formula Revision Guide