Rearranging Harder Formulae

What you'll learn

How to make a chosen letter the subject of a formula.
How to undo brackets, powers and fractions in the right order.
How to handle formulae where the subject appears more than once.
How to spot common traps in Grade 7 rearranging questions.

1. Start with the subject

When you rearrange a formula, your aim is to get one chosen letter on its own.

All the other letters are treated like numbers. For example, if you are making $x$ the subject, then $a$ , $b$ and $c$ are just constants.

Definition

Subject of a formula

The subject of a formula is the letter that is on its own on one side of the equals sign. In $\pi r^2$ , the subject is $A$ .

The subject of a formula is the single letter isolated on one side of the equals sign.

Key Idea

The balance rule

Whatever you do to one side of the equation, you must do to the other side as well.

An equation stays balanced only when the same operation is applied to both sides.

Example

Make $r$ the subject of $w = r + k t$

To make r the subject, undo the added kt by subtracting kt from both sides.

Start with the formula.
$w = r + k t$
The term $k t$ is being added to $r$ , so subtract $k t$ from both sides.
$w - k t = r$
Write the answer with $r$ on the left.
$r = w - k t$

2. Use inverse operations

An inverse operation is the operation that undoes another operation. Adding is undone by subtracting. Multiplying is undone by dividing. Squaring is undone by square rooting.

If the subject is inside several operations, undo them from the outside in.

Example

Make $k$ the subject of $w = r + k t$

To isolate k, undo the outside addition first, then undo multiplication by t.

Start with the formula.
$w = r + k t$
Subtract $r$ from both sides to isolate the term containing $k$ .
$w - r = k t$
Divide both sides by $t$ .
$\frac{w - r}{t}$

Tip

Treat letters like numbers

If you are making $k$ the subject, then $r$ and $t$ behave like fixed numbers. Do not be put off just because they are letters.

3. Rearranging with squares and roots

A square root undoes a square. If you have $p^2$ , square rooting gives $p$ .

Be careful: in pure algebra, square rooting can give a positive or negative answer.

Common Mistake

Square roots can have two signs

If the formula is about a length, speed or other quantity that cannot be negative, you usually take the positive square root. Without context, write $±\pm$ .

Example

Make $p$ the subject of $q^{2} = p^{2} + 5 r s$

First isolate p², then square root both sides to make p the subject.

Start by isolating $p^2$ . Subtract $5 r s$ from both sides.
$q^2 - 5rs = p^2$
Square root both sides.
$\pm\sqrt{q^2 - 5rs}$

4. Fractions attached to terms

A coefficient is a number or expression multiplying a variable. In $12kt2\frac{1}{2}kt^2$ , the coefficient of $k$ is $12t2\frac{1}{2}t^2$ .

To remove a fraction like $12\frac{1}{2}$ , multiply by 2.

Example

Make $k$ the subject of $D = n t + \frac{1}{2} k t^{2}$

Subtract $n t$ from both sides.
$\frac{1}{2}kt^2$
Multiply both sides by 2.
$2(D - nt) = kt^2$
Divide both sides by $t^2$ .
$\frac{2(D - nt)}{t^2}$

Common Mistake

Forgetting the whole coefficient

In $12kt2\frac{1}{2}kt^2$ , the letter $k$ is multiplied by both $12\frac{1}{2}$ and $t^2$ . You must undo both parts.

The whole coefficient attached to k is both 1/2 and t², so both must be undone.

5. When the subject appears more than once

Sometimes the letter you want appears in two places. You cannot make it the subject until you have collected all those terms together.

To factorise means to take out a common factor. For example, $4 y + cy - d y$ becomes $y (4 + c - d)$ .

Factorising collects repeated y terms by taking y outside a bracket.

Example

Make $y$ the subject of $y (4 + c) = d (y + 5)$

Expand the brackets.
$4 y + cy = d y + 5 d$
Move all the $y$ terms to one side.
$4 y + cy - d y = 5 d$
Factorise by taking out $y$ .
$y (4 + c - d) = 5 d$
Divide by $4 + c - d$ .
$\frac{5d}{4 + c - d}$

Common Mistake

Cancelling too early

Do not cancel the subject letter from both sides if it appears in separate added or subtracted terms. Collect and factorise first.

6. Algebraic fractions

A denominator is the bottom part of a fraction. In $x+1x−3\frac{x + 1}{x - 3}$ , the denominator is $x - 3$ .

To remove an algebraic fraction, multiply both sides by the denominator. This is often called clearing the fraction.

Example

Make $y$ the subject of $a = \frac{5 + 3 cy}{3 y - 2}$

Multiply both sides by $3 y - 2$ .
$a (3 y - 2) = 5 + 3 cy$
Expand the left-hand side.
$3 a y - 2 a = 5 + 3 cy$
Collect the $y$ terms on one side.
$3 a y - 3 cy = 5 + 2 a$
Factorise and divide.
$y(3a−3c)=5+2ay=5+2a3a−3c\begin{aligned} y(3a - 3c) &= 5 + 2a \\ y &= \frac{5 + 2a}{3a - 3c} \end{aligned}$

Tip

Check denominators

Original denominators cannot be zero. In the example above, $\neq 0$ , so any final answer must respect that restriction.

7. Reciprocal formulae

The reciprocal of a number or expression is 1 divided by it. For example, the reciprocal of $q$ is $1q\frac{1}{q}$ .

For reciprocal formulae, a reliable method is to multiply every term by all the denominators.

Multiplying every term by pqr clears the reciprocal denominators p, q and r.

Example

Make $q$ the subject of $\frac{1}{p} = \frac{1}{q} + \frac{1}{r}$

Start with the formula.
$1p=1q+1r\frac{1}{p} = \frac{1}{q} + \frac{1}{r}$
Multiply every term by $pq r$ .
$q r = p r + pq$
Collect the terms containing $q$ .
$q r - pq = p r$
Factorise and divide.
$q(r−p)=prq=prr−p\begin{aligned} q(r - p) &= pr \\ q &= \frac{pr}{r - p} \end{aligned}$

Exam technique

In the exam

Circle the letter you are making the subject before you start.
Clear brackets and fractions early if they are blocking the subject.
If the subject appears more than once, collect those terms, factorise, then divide.
Check your answer by seeing whether the subject is completely alone on one side.

Self review

Check yourself

Can you explain why $x (a + b)$ cannot usually be changed into $x a + b$ ?
If $x$ appears on both sides of a formula, what three actions should you try?
When rearranging a formula with $v^2$ , when might you need a $±\pm$ sign?

Rearranging Harder Formulae

What you'll learn

How to make a chosen letter the subject of a formula.
How to undo brackets, powers and fractions in the right order.
How to handle formulae where the subject appears more than once.
How to spot common traps in Grade 7 rearranging questions.

1. Start with the subject

When you rearrange a formula, your aim is to get one chosen letter on its own.

All the other letters are treated like numbers. For example, if you are making $x$ the subject, then $a$ , $b$ and $c$ are just constants.

Definition

Subject of a formula

The subject of a formula is the letter that is on its own on one side of the equals sign. In $\pi r^2$ , the subject is $A$ .

The subject of a formula is the single letter isolated on one side of the equals sign.

Key Idea

The balance rule

Whatever you do to one side of the equation, you must do to the other side as well.

An equation stays balanced only when the same operation is applied to both sides.

Example

Make $r$ the subject of $w = r + k t$

To make r the subject, undo the added kt by subtracting kt from both sides.

Start with the formula.
$w = r + k t$
The term $k t$ is being added to $r$ , so subtract $k t$ from both sides.
$w - k t = r$
Write the answer with $r$ on the left.
$r = w - k t$

2. Use inverse operations

An inverse operation is the operation that undoes another operation. Adding is undone by subtracting. Multiplying is undone by dividing. Squaring is undone by square rooting.

If the subject is inside several operations, undo them from the outside in.

Example

Make $k$ the subject of $w = r + k t$

To isolate k, undo the outside addition first, then undo multiplication by t.

Start with the formula.
$w = r + k t$
Subtract $r$ from both sides to isolate the term containing $k$ .
$w - r = k t$
Divide both sides by $t$ .
$\frac{w - r}{t}$

Tip

Treat letters like numbers

If you are making $k$ the subject, then $r$ and $t$ behave like fixed numbers. Do not be put off just because they are letters.

3. Rearranging with squares and roots

A square root undoes a square. If you have $p^2$ , square rooting gives $p$ .

Be careful: in pure algebra, square rooting can give a positive or negative answer.

Common Mistake

Square roots can have two signs

If the formula is about a length, speed or other quantity that cannot be negative, you usually take the positive square root. Without context, write $±\pm$ .

Example

Make $p$ the subject of $q^{2} = p^{2} + 5 r s$

First isolate p², then square root both sides to make p the subject.

Start by isolating $p^2$ . Subtract $5 r s$ from both sides.
$q^2 - 5rs = p^2$
Square root both sides.
$\pm\sqrt{q^2 - 5rs}$

4. Fractions attached to terms

A coefficient is a number or expression multiplying a variable. In $12kt2\frac{1}{2}kt^2$ , the coefficient of $k$ is $12t2\frac{1}{2}t^2$ .

To remove a fraction like $12\frac{1}{2}$ , multiply by 2.

Example

Make $k$ the subject of $D = n t + \frac{1}{2} k t^{2}$

Subtract $n t$ from both sides.
$\frac{1}{2}kt^2$
Multiply both sides by 2.
$2(D - nt) = kt^2$
Divide both sides by $t^2$ .
$\frac{2(D - nt)}{t^2}$

Common Mistake

Forgetting the whole coefficient

In $12kt2\frac{1}{2}kt^2$ , the letter $k$ is multiplied by both $12\frac{1}{2}$ and $t^2$ . You must undo both parts.

The whole coefficient attached to k is both 1/2 and t², so both must be undone.

5. When the subject appears more than once

Sometimes the letter you want appears in two places. You cannot make it the subject until you have collected all those terms together.

To factorise means to take out a common factor. For example, $4 y + cy - d y$ becomes $y (4 + c - d)$ .

Factorising collects repeated y terms by taking y outside a bracket.

Example

Make $y$ the subject of $y (4 + c) = d (y + 5)$

Expand the brackets.
$4 y + cy = d y + 5 d$
Move all the $y$ terms to one side.
$4 y + cy - d y = 5 d$
Factorise by taking out $y$ .
$y (4 + c - d) = 5 d$
Divide by $4 + c - d$ .
$\frac{5d}{4 + c - d}$

Common Mistake

Cancelling too early

Do not cancel the subject letter from both sides if it appears in separate added or subtracted terms. Collect and factorise first.

6. Algebraic fractions

A denominator is the bottom part of a fraction. In $x+1x−3\frac{x + 1}{x - 3}$ , the denominator is $x - 3$ .

To remove an algebraic fraction, multiply both sides by the denominator. This is often called clearing the fraction.

Example

Make $y$ the subject of $a = \frac{5 + 3 cy}{3 y - 2}$

Multiply both sides by $3 y - 2$ .
$a (3 y - 2) = 5 + 3 cy$
Expand the left-hand side.
$3 a y - 2 a = 5 + 3 cy$
Collect the $y$ terms on one side.
$3 a y - 3 cy = 5 + 2 a$
Factorise and divide.
$y(3a−3c)=5+2ay=5+2a3a−3c\begin{aligned} y(3a - 3c) &= 5 + 2a \\ y &= \frac{5 + 2a}{3a - 3c} \end{aligned}$

Tip

Check denominators

Original denominators cannot be zero. In the example above, $\neq 0$ , so any final answer must respect that restriction.

7. Reciprocal formulae

The reciprocal of a number or expression is 1 divided by it. For example, the reciprocal of $q$ is $1q\frac{1}{q}$ .

For reciprocal formulae, a reliable method is to multiply every term by all the denominators.

Multiplying every term by pqr clears the reciprocal denominators p, q and r.

Example

Make $q$ the subject of $\frac{1}{p} = \frac{1}{q} + \frac{1}{r}$

Start with the formula.
$1p=1q+1r\frac{1}{p} = \frac{1}{q} + \frac{1}{r}$
Multiply every term by $pq r$ .
$q r = p r + pq$
Collect the terms containing $q$ .
$q r - pq = p r$
Factorise and divide.
$q(r−p)=prq=prr−p\begin{aligned} q(r - p) &= pr \\ q &= \frac{pr}{r - p} \end{aligned}$

Exam technique

In the exam

Circle the letter you are making the subject before you start.
Clear brackets and fractions early if they are blocking the subject.
If the subject appears more than once, collect those terms, factorise, then divide.
Check your answer by seeing whether the subject is completely alone on one side.

Self review

Check yourself

Can you explain why $x (a + b)$ cannot usually be changed into $x a + b$ ?
If $x$ appears on both sides of a formula, what three actions should you try?
When rearranging a formula with $v^2$ , when might you need a $±\pm$ sign?

Rearranging Harder Formulae

What you'll learn

1. Start with the subject

2. Use inverse operations

3. Rearranging with squares and roots

4. Fractions attached to terms

5. When the subject appears more than once

6. Algebraic fractions

7. Reciprocal formulae

Recap questions

You've reached the end

Rearranging Harder Formulae

What you'll learn

1. Start with the subject

2. Use inverse operations

3. Rearranging with squares and roots

4. Fractions attached to terms

5. When the subject appears more than once

6. Algebraic fractions

7. Reciprocal formulae

Recap questions

You've reached the end

Rearranging Harder Formulae

What you'll learn

1. Start with the subject

2. Use inverse operations

3. Rearranging with squares and roots

4. Fractions attached to terms

5. When the subject appears more than once

6. Algebraic fractions

7. Reciprocal formulae

Recap questions

You've reached the end

Rearranging Harder Formulae Revision Guide

Rearranging Harder Formulae

What you'll learn

1. Start with the subject

2. Use inverse operations

3. Rearranging with squares and roots

4. Fractions attached to terms

5. When the subject appears more than once

6. Algebraic fractions

7. Reciprocal formulae

Recap questions

You've reached the end

Rearranging Harder Formulae Revision Guide