Rearranging Harder Formulae

What you'll learn

How to make a chosen letter the subject of a formula.
How to undo brackets, fractions, squares and square roots.
How to rearrange formulae where the subject appears more than once.
How to handle reciprocal formulae such as $1a=1b+1c\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$ .

The goal: make one variable the subject

A formula is an equation linking variables. A variable is a letter that represents a quantity, such as time, speed, mass or length.

Definition

Subject of a formula

The subject of a formula is the variable written on its own on one side. For example, in $v = u + a t$ , the subject is $v$ .

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

xx=\text{something not containing }x

Harder rearranging usually means the target letter is trapped inside brackets, fractions, powers, or appears in more than one place.

The balance method and inverse operations

Every rearrangement uses the same principle: whatever you do to one side, you must do to the other side.

An inverse operation is an operation that undoes another one. Subtracting undoes adding, dividing undoes multiplying, and square-rooting undoes squaring. Rearranging is often about undoing operations in the reverse order.

Flow diagram showing forward operations and reverse inverse operations for rearranging a formula

Key Idea

Main idea

To rearrange a formula, keep the equation balanced and work backwards until the target letter is alone.

Example

Making a letter the subject in a linear formula

Make $k$ the subject of $m = n + k p$ .

The target is $k$ . First remove $n$ by subtracting $n$ from both sides:
$m - n = k p$
Now $k$ is multiplied by $p$ , so divide both sides by $p$ :
$k=m−npk=\frac{m-n}{p}$

Common Mistake

Do not just 'move' terms

It is safer to think “do the opposite operation to both sides” rather than “move it across”. This helps you avoid sign errors.

Rearranging when there is a square

If the target letter is squared, isolate the squared part first, then take the square root.

The square root is the inverse of squaring. For example, if $x^2=25$ , then $x=±5x=\pm5$ because both positive and negative values square to 25.

Example

Making a squared letter the subject

Make $P$ the subject of $R^2=P^2+5qt$ .

The target is $P$ , but it is currently inside $P^2$ . First subtract $5 q t$ from both sides:
$R^2-5qt=P^2$
Take the square root of both sides:
$P=±R2−5qtP=\pm\sqrt{R^2-5qt}$

Tip

When to use the positive root only

In a real-life context, if the variable represents a length, speed or time, the negative answer may not make sense. In pure algebra, include $±\pm$ unless the question gives a reason not to.

Clearing fractions and halves

A denominator is the bottom part of a fraction. To remove a denominator, multiply both sides by that denominator.

This is especially useful when a formula contains $12\frac{1}{2}$ .

Example

Removing a half-term

Make $n$ the subject of $d=wt+12nt2d=wt+\frac{1}{2}nt^2$ .

Remove the term not containing $n$ by subtracting $w t$ from both sides:
$d−wt=12nt2d-wt=\frac{1}{2}nt^2$
Multiply both sides by 2 to remove the half:
$2(d-wt)=nt^2$
Divide both sides by $t^2$ :
$n=2(d−wt)t2n=\frac{2(d-wt)}{t^2}$

Common Mistake

Forgetting the whole term

In $12nt2\frac{1}{2}nt^2$ , the half applies to the whole product $nt^2$ . Remove the half before dividing by $t^2$ .

When the subject appears more than once

Sometimes the letter you want appears in two places. Your aim is to collect all the target-letter terms on one side, then factorise.

Definition

Factorising

Factorising means taking out a common factor. For example, $a x + b x = x (a + b)$ because both terms contain $x$ .

A coefficient is the number or algebraic factor multiplying a variable. In $7 x$ , the coefficient of $x$ is 7. In $a x$ , the coefficient of $x$ is $a$ .

Example

Collecting and factorising the target letter

Make $x$ the subject of $x (4 + a) = b (x + 1)$ .

Expand both brackets:
$4 x + a x = b x + b$
Collect the $x$ terms on the left by subtracting $b x$ from both sides:
$4 x + a x - b x = b$
Factorise the left-hand side by taking out $x$ :
$x (4 + a - b) = b$
Divide by the bracket multiplying $x$ :
$x=b4+a−bx=\frac{b}{4+a-b}$

Tip

Collect, factorise, divide

If the target appears more than once, use this routine: collect the target terms, factorise the target letter, then divide by what is left.

Algebraic fractions with the subject in the numerator and denominator

An algebraic fraction is a fraction that contains variables. If the subject is in a denominator, clear the fraction first by multiplying both sides by the denominator.

Example

Clearing an algebraic fraction

Make $x$ the subject of $p=6+2qx2x−5p=\frac{6+2qx}{2x-5}$ .

Multiply both sides by $2 x - 5$ :
$p (2 x - 5) = 6 + 2 q x$
Expand the left-hand side:
$2 p x - 5 p = 6 + 2 q x$
Collect the $x$ terms on the left and the non- $x$ terms on the right:
$2 p x - 2 q x = 6 + 5 p$
Factorise the left-hand side:
$2 x (p - q) = 6 + 5 p$
Divide by $2 (p - q)$ :
$x=6+5p2(p−q)x=\frac{6+5p}{2(p-q)}$

Common Mistake

Hidden restrictions

You cannot divide by zero. In rearranging questions, assume denominators such as $2 x - 5$ and final divisors such as $p - q$ are non-zero unless a special case is being considered.

Fractions equal to fractions

When you have a fraction equal to a fraction, you can often cross-multiply. This means multiplying both sides by both denominators to remove the fractions.

Example

Using cross-multiplication

Make $x$ the subject of $ab=3xx+4\frac{a}{b}=\frac{3x}{x+4}$ .

Multiply both sides by $b (x + 4)$ :
$a (x + 4) = 3 b x$
Expand the bracket:
$a x + 4 a = 3 b x$
Collect the $x$ terms on the right:
$4 a = 3 b x - a x$
Factorise the right-hand side:
$4 a = x (3 b - a)$
Divide by $3 b - a$ :
$x=4a3b−ax=\frac{4a}{3b-a}$

Reciprocal formulae

A reciprocal means “one over” a quantity. For example, the reciprocal of $y$ is $1y\frac{1}{y}$ , as long as $y≠0y\neq0$ .

Reciprocal formulae look awkward, but the method is usually:

isolate the reciprocal containing the target;
combine the other fractions;
take the reciprocal of both sides.

Example

Making a variable the subject in a reciprocal formula

Make $y$ the subject of $1x=1y+1z\frac{1}{x}=\frac{1}{y}+\frac{1}{z}$ .

Isolate $1y\frac{1}{y}$ by subtracting $1z\frac{1}{z}$ from both sides:
$1y=1x−1z\frac{1}{y}=\frac{1}{x}-\frac{1}{z}$
Combine the fractions on the right using common denominator $x z$ :
$1y=z−xxz\frac{1}{y}=\frac{z-x}{xz}$
Take the reciprocal of both sides:
$y=xzz−xy=\frac{xz}{z-x}$

Exam technique

In the exam

Identify the target letter first, then underline every place it appears.
Clear fractions and expand brackets before trying to collect terms.
If the target appears more than once, collect, factorise, then divide.
Check your final answer: the subject should appear once, on its own, and not on the other side.

Self review

Check yourself

If the target letter appears in two separate terms, what is the key step before dividing?
When you take the square root of both sides, when might you need a $±\pm$ sign?
Why is it helpful to clear denominators before expanding brackets?

Rearranging Harder Formulae

What you'll learn

How to make a chosen letter the subject of a formula.
How to undo brackets, fractions, squares and square roots.
How to rearrange formulae where the subject appears more than once.
How to handle reciprocal formulae such as $1a=1b+1c\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$ .

The goal: make one variable the subject

A formula is an equation linking variables. A variable is a letter that represents a quantity, such as time, speed, mass or length.

Definition

Subject of a formula

The subject of a formula is the variable written on its own on one side. For example, in $v = u + a t$ , the subject is $v$ .

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

xx=\text{something not containing }x

Harder rearranging usually means the target letter is trapped inside brackets, fractions, powers, or appears in more than one place.

The balance method and inverse operations

Every rearrangement uses the same principle: whatever you do to one side, you must do to the other side.

Flow diagram showing forward operations and reverse inverse operations for rearranging a formula

Key Idea

Main idea

To rearrange a formula, keep the equation balanced and work backwards until the target letter is alone.

Example

Making a letter the subject in a linear formula

Make $k$ the subject of $m = n + k p$ .

The target is $k$ . First remove $n$ by subtracting $n$ from both sides:
$m - n = k p$
Now $k$ is multiplied by $p$ , so divide both sides by $p$ :
$k=m−npk=\frac{m-n}{p}$

Common Mistake

Do not just 'move' terms

It is safer to think “do the opposite operation to both sides” rather than “move it across”. This helps you avoid sign errors.

Rearranging when there is a square

If the target letter is squared, isolate the squared part first, then take the square root.

The square root is the inverse of squaring. For example, if $x^2=25$ , then $x=±5x=\pm5$ because both positive and negative values square to 25.

Example

Making a squared letter the subject

Make $P$ the subject of $R^2=P^2+5qt$ .

The target is $P$ , but it is currently inside $P^2$ . First subtract $5 q t$ from both sides:
$R^2-5qt=P^2$
Take the square root of both sides:
$P=±R2−5qtP=\pm\sqrt{R^2-5qt}$

Tip

When to use the positive root only

In a real-life context, if the variable represents a length, speed or time, the negative answer may not make sense. In pure algebra, include $±\pm$ unless the question gives a reason not to.

Clearing fractions and halves

A denominator is the bottom part of a fraction. To remove a denominator, multiply both sides by that denominator.

This is especially useful when a formula contains $12\frac{1}{2}$ .

Example

Removing a half-term

Make $n$ the subject of $d=wt+12nt2d=wt+\frac{1}{2}nt^2$ .

Remove the term not containing $n$ by subtracting $w t$ from both sides:
$d−wt=12nt2d-wt=\frac{1}{2}nt^2$
Multiply both sides by 2 to remove the half:
$2(d-wt)=nt^2$
Divide both sides by $t^2$ :
$n=2(d−wt)t2n=\frac{2(d-wt)}{t^2}$

Common Mistake

Forgetting the whole term

In $12nt2\frac{1}{2}nt^2$ , the half applies to the whole product $nt^2$ . Remove the half before dividing by $t^2$ .

When the subject appears more than once

Sometimes the letter you want appears in two places. Your aim is to collect all the target-letter terms on one side, then factorise.

Definition

Factorising

Factorising means taking out a common factor. For example, $a x + b x = x (a + b)$ because both terms contain $x$ .

A coefficient is the number or algebraic factor multiplying a variable. In $7 x$ , the coefficient of $x$ is 7. In $a x$ , the coefficient of $x$ is $a$ .

Example

Collecting and factorising the target letter

Make $x$ the subject of $x (4 + a) = b (x + 1)$ .

Expand both brackets:
$4 x + a x = b x + b$
Collect the $x$ terms on the left by subtracting $b x$ from both sides:
$4 x + a x - b x = b$
Factorise the left-hand side by taking out $x$ :
$x (4 + a - b) = b$
Divide by the bracket multiplying $x$ :
$x=b4+a−bx=\frac{b}{4+a-b}$

Tip

Collect, factorise, divide

If the target appears more than once, use this routine: collect the target terms, factorise the target letter, then divide by what is left.

Algebraic fractions with the subject in the numerator and denominator

An algebraic fraction is a fraction that contains variables. If the subject is in a denominator, clear the fraction first by multiplying both sides by the denominator.

Example

Clearing an algebraic fraction

Make $x$ the subject of $p=6+2qx2x−5p=\frac{6+2qx}{2x-5}$ .

Multiply both sides by $2 x - 5$ :
$p (2 x - 5) = 6 + 2 q x$
Expand the left-hand side:
$2 p x - 5 p = 6 + 2 q x$
Collect the $x$ terms on the left and the non- $x$ terms on the right:
$2 p x - 2 q x = 6 + 5 p$
Factorise the left-hand side:
$2 x (p - q) = 6 + 5 p$
Divide by $2 (p - q)$ :
$x=6+5p2(p−q)x=\frac{6+5p}{2(p-q)}$

Common Mistake

Hidden restrictions

You cannot divide by zero. In rearranging questions, assume denominators such as $2 x - 5$ and final divisors such as $p - q$ are non-zero unless a special case is being considered.

Fractions equal to fractions

When you have a fraction equal to a fraction, you can often cross-multiply. This means multiplying both sides by both denominators to remove the fractions.

Example

Using cross-multiplication

Make $x$ the subject of $ab=3xx+4\frac{a}{b}=\frac{3x}{x+4}$ .

Multiply both sides by $b (x + 4)$ :
$a (x + 4) = 3 b x$
Expand the bracket:
$a x + 4 a = 3 b x$
Collect the $x$ terms on the right:
$4 a = 3 b x - a x$
Factorise the right-hand side:
$4 a = x (3 b - a)$
Divide by $3 b - a$ :
$x=4a3b−ax=\frac{4a}{3b-a}$

Reciprocal formulae

A reciprocal means “one over” a quantity. For example, the reciprocal of $y$ is $1y\frac{1}{y}$ , as long as $y≠0y\neq0$ .

Reciprocal formulae look awkward, but the method is usually:

isolate the reciprocal containing the target;
combine the other fractions;
take the reciprocal of both sides.

Example

Making a variable the subject in a reciprocal formula

Make $y$ the subject of $1x=1y+1z\frac{1}{x}=\frac{1}{y}+\frac{1}{z}$ .

Isolate $1y\frac{1}{y}$ by subtracting $1z\frac{1}{z}$ from both sides:
$1y=1x−1z\frac{1}{y}=\frac{1}{x}-\frac{1}{z}$
Combine the fractions on the right using common denominator $x z$ :
$1y=z−xxz\frac{1}{y}=\frac{z-x}{xz}$
Take the reciprocal of both sides:
$y=xzz−xy=\frac{xz}{z-x}$

Exam technique

In the exam

Identify the target letter first, then underline every place it appears.
Clear fractions and expand brackets before trying to collect terms.
If the target appears more than once, collect, factorise, then divide.
Check your final answer: the subject should appear once, on its own, and not on the other side.

Self review

Check yourself

If the target letter appears in two separate terms, what is the key step before dividing?
When you take the square root of both sides, when might you need a $±\pm$ sign?
Why is it helpful to clear denominators before expanding brackets?

Rearranging Harder Formulae

What you'll learn

The goal: make one variable the subject

The balance method and inverse operations

Rearranging when there is a square

Clearing fractions and halves

When the subject appears more than once

Algebraic fractions with the subject in the numerator and denominator

Fractions equal to fractions

Reciprocal formulae

Recap questions

You've reached the end

Rearranging Harder Formulae

What you'll learn

The goal: make one variable the subject

The balance method and inverse operations

Rearranging when there is a square

Clearing fractions and halves

When the subject appears more than once

Algebraic fractions with the subject in the numerator and denominator

Fractions equal to fractions

Reciprocal formulae

Recap questions

You've reached the end

Rearranging Harder Formulae

What you'll learn

The goal: make one variable the subject

The balance method and inverse operations

Rearranging when there is a square

Clearing fractions and halves

When the subject appears more than once

Algebraic fractions with the subject in the numerator and denominator

Fractions equal to fractions

Reciprocal formulae

Recap questions

You've reached the end

Rearranging Harder Formulae Revision Guide

Rearranging Harder Formulae

What you'll learn

The goal: make one variable the subject

The balance method and inverse operations

Rearranging when there is a square

Clearing fractions and halves

When the subject appears more than once

Algebraic fractions with the subject in the numerator and denominator

Fractions equal to fractions

Reciprocal formulae

Recap questions

You've reached the end

Rearranging Harder Formulae Revision Guide