Solving One Step Equations

What you'll learn

How to read a calculation with a missing number.
How boxes and letters can stand for a missing value.
How to undo addition, subtraction, multiplication and division.
How to check your answer quickly.

1. The equals sign means both sides match

Before using letters, start with missing-number questions. The equals sign means the value on the left is the same as the value on the right.

Definition

Key words

An equation is a mathematical sentence with an equals sign.
A variable is a letter, such as $x$ or $m$ , that stands for a number you do not know yet.
An operation is a calculation action, such as add, subtract, multiply or divide.
An inverse operation is the opposite operation that undoes another one. Addition and subtraction undo each other; multiplication and division undo each other.
To solve an equation means to find the value of the variable that makes the equation true.
A one-step equation is solved using just one inverse operation.

Key Idea

Equals means balance

The left side and the right side of an equation must have the same value. Your answer is correct only if it makes the whole calculation true.

Example

A missing number after addition

Find the missing number in $6+00=176+\boxed{\phantom{00}}=17$ .

A bar model shows that the missing add-on is the gap from 6 up to 17.

The left side already has 6.
It must reach 17, so find the gap: 17 - 6 = 11.
The missing number is 11, because 6 + 11 = 17.

When the missing number is being subtracted

Sometimes the missing number is the amount being taken away.

Example

A missing number being subtracted

Find the missing number in $19−00=1219-\boxed{\phantom{00}}=12$ .

A take-away bar model shows that 19 is split into the remaining 12 and the missing amount removed.

Read it as “19 subtract something equals 12.”
Work out how much has been taken away: 19 - 12 = 7.
The missing number is 7, because 19 - 7 = 12.

2. Replacing the box with a letter

A letter works just like a box. For example, $x + 7 = 19$ means “a number plus 7 equals 19”.

Your aim is to get the letter on its own.

Key Idea

Undo to solve

To solve a one-step equation, undo the operation next to the variable.

Example

Solving an addition equation

Solve $x + 7 = 19$ .

The balance diagram represents subtracting 7 from both sides to leave x on its own.

The operation on $x$ is add 7.
The inverse operation is subtract 7.
Work out 19 - 7 = 12, so $x = 12$ .
Check: 12 + 7 = 19.

3. Undoing subtraction

There are two common subtraction patterns.

If the variable has a number subtracted from it, undo this by adding.

Example

Letter minus a number

Solve $m - 6 = 10$ .

$m$ has had 6 subtracted from it.
Undo subtract 6 by adding 6.
Work out 10 + 6 = 16, so $m = 16$ .
Check: 16 - 6 = 10.

If the variable is being subtracted from a number, find what was taken away. The difference is the gap between two numbers.

Example

Number minus a letter

Solve $23 - n = 14$ .

The bar model makes clear that n is the amount taken away from 23, not 14 minus 23.

Read it as “23 subtract $n$ equals 14.”
Find what was taken away: 23 - 14 = 9.
So $n = 9$ .
Check: 23 - 9 = 14.

Common Mistake

Subtracting in the wrong order

For $23 - n = 14$ , do not work out 14 - 23. The letter is the amount taken away from 23, so find 23 - 14.

4. Undoing multiplication

When a number is written next to a letter, it means multiply.

Definition

Multiplication shorthand

When a number is written next to a variable, such as $5 p$ , it means $\times p$ .

To undo multiplying, divide.

Example

Solving a multiplication equation

Solve $7 a = 42$ .

Seven equal groups of a make 42, so each group is found by sharing 42 equally into 7 parts.

$7 a$ means $\times a$ .
Undo multiplying by 7 by dividing by 7.
Work out 42 divided by 7 = 6, so $a = 6$ .
Check: 7 times 6 is 42.

Repeated letters also mean multiplication.

Example

Repeated letters

Solve $r + r + r = 27$ .

Three equal parts labelled r add together to make a total of 27.

There are 3 copies of $r$ , so $r + r + r$ is the same as $3 r$ .
Solve $3 r = 27$ by dividing by 3.
Work out 27 divided by 3 = 9, so $r = 9$ .
Check: 9 + 9 + 9 = 27.

5. Undoing division and fractions

The line in a fraction is called the fraction bar. It means divide. So $d2\frac{d}{2}$ means $d$ divided by 2.

To undo dividing, multiply.

Example

Letter divided by a number

Solve $d2=7.5\frac{d}{2}=7.5$ .

The diagram shows d divided into two equal halves, with each half worth 7.5.

This means $d$ divided by 2 equals 7.5.
Undo dividing by 2 by multiplying by 2.
Work out 7.5 times 2 = 15, so $d = 15$ .
Check: 15 divided by 2 is 7.5.

Sometimes the letter is on the bottom of the fraction.

Definition

Denominator

The denominator is the bottom part of a fraction. In $30y\frac{30}{y}$ , the denominator is $y$ .

Example

Letter on the bottom

Solve $30y=6\frac{30}{y}=6$ .

Thirty divided into y equal groups gives groups of 6, so y is the number of equal groups.

Read it as “30 divided by $y$ equals 6.”
Ask: “30 divided by what gives 6?”
Work out 30 divided by 6 = 5, so $y = 5$ .
Check: 30 divided by 5 is 6.

Common Mistake

Letter on the bottom

For $30y=6\frac{30}{y}=6$ , the answer is not 30 times 6. Ask “30 divided by what is 6?” instead.

6. Checking your answer

Checking is one of the best ways to avoid losing easy marks.

Definition

Substitute

To substitute means to replace a variable with a number.

Example

Solve then check

Solve $5 + p = 13$ .

Addition can be in either order, so read this as $p + 5 = 13$ .
Undo add 5 by subtracting 5.
Work out 13 - 5 = 8, so $p = 8$ .
Substitute 8 back in: 5 + 8 = 13, so the answer works.

Exam technique

In the exam

Look for the operation attached to the letter: add, subtract, multiply or divide.
Use the inverse operation to get the letter on its own.
Check by putting your answer back into the original equation.

Self review

Check yourself

If $x + 7 = 20$ , what operation undoes add 7?
What does $6 a$ mean in words?
Why is $18 - y = 11$ different from $y - 18 = 11$ ?

Solving One Step Equations

What you'll learn

How to read a calculation with a missing number.
How boxes and letters can stand for a missing value.
How to undo addition, subtraction, multiplication and division.
How to check your answer quickly.

1. The equals sign means both sides match

Before using letters, start with missing-number questions. The equals sign means the value on the left is the same as the value on the right.

Definition

Key words

An equation is a mathematical sentence with an equals sign.
A variable is a letter, such as $x$ or $m$ , that stands for a number you do not know yet.
An operation is a calculation action, such as add, subtract, multiply or divide.
An inverse operation is the opposite operation that undoes another one. Addition and subtraction undo each other; multiplication and division undo each other.
To solve an equation means to find the value of the variable that makes the equation true.
A one-step equation is solved using just one inverse operation.

Key Idea

Equals means balance

The left side and the right side of an equation must have the same value. Your answer is correct only if it makes the whole calculation true.

Example

A missing number after addition

Find the missing number in $6+00=176+\boxed{\phantom{00}}=17$ .

A bar model shows that the missing add-on is the gap from 6 up to 17.

The left side already has 6.
It must reach 17, so find the gap: 17 - 6 = 11.
The missing number is 11, because 6 + 11 = 17.

When the missing number is being subtracted

Sometimes the missing number is the amount being taken away.

Example

A missing number being subtracted

Find the missing number in $19−00=1219-\boxed{\phantom{00}}=12$ .

A take-away bar model shows that 19 is split into the remaining 12 and the missing amount removed.

Read it as “19 subtract something equals 12.”
Work out how much has been taken away: 19 - 12 = 7.
The missing number is 7, because 19 - 7 = 12.

2. Replacing the box with a letter

A letter works just like a box. For example, $x + 7 = 19$ means “a number plus 7 equals 19”.

Your aim is to get the letter on its own.

Key Idea

Undo to solve

To solve a one-step equation, undo the operation next to the variable.

Example

Solving an addition equation

Solve $x + 7 = 19$ .

The balance diagram represents subtracting 7 from both sides to leave x on its own.

The operation on $x$ is add 7.
The inverse operation is subtract 7.
Work out 19 - 7 = 12, so $x = 12$ .
Check: 12 + 7 = 19.

3. Undoing subtraction

There are two common subtraction patterns.

If the variable has a number subtracted from it, undo this by adding.

Example

Letter minus a number

Solve $m - 6 = 10$ .

$m$ has had 6 subtracted from it.
Undo subtract 6 by adding 6.
Work out 10 + 6 = 16, so $m = 16$ .
Check: 16 - 6 = 10.

If the variable is being subtracted from a number, find what was taken away. The difference is the gap between two numbers.

Example

Number minus a letter

Solve $23 - n = 14$ .

The bar model makes clear that n is the amount taken away from 23, not 14 minus 23.

Read it as “23 subtract $n$ equals 14.”
Find what was taken away: 23 - 14 = 9.
So $n = 9$ .
Check: 23 - 9 = 14.

Common Mistake

Subtracting in the wrong order

For $23 - n = 14$ , do not work out 14 - 23. The letter is the amount taken away from 23, so find 23 - 14.

4. Undoing multiplication

When a number is written next to a letter, it means multiply.

Definition

Multiplication shorthand

When a number is written next to a variable, such as $5 p$ , it means $\times p$ .

To undo multiplying, divide.

Example

Solving a multiplication equation

Solve $7 a = 42$ .

Seven equal groups of a make 42, so each group is found by sharing 42 equally into 7 parts.

$7 a$ means $\times a$ .
Undo multiplying by 7 by dividing by 7.
Work out 42 divided by 7 = 6, so $a = 6$ .
Check: 7 times 6 is 42.

Repeated letters also mean multiplication.

Example

Repeated letters

Solve $r + r + r = 27$ .

Three equal parts labelled r add together to make a total of 27.

There are 3 copies of $r$ , so $r + r + r$ is the same as $3 r$ .
Solve $3 r = 27$ by dividing by 3.
Work out 27 divided by 3 = 9, so $r = 9$ .
Check: 9 + 9 + 9 = 27.

5. Undoing division and fractions

The line in a fraction is called the fraction bar. It means divide. So $d2\frac{d}{2}$ means $d$ divided by 2.

To undo dividing, multiply.

Example

Letter divided by a number

Solve $d2=7.5\frac{d}{2}=7.5$ .

The diagram shows d divided into two equal halves, with each half worth 7.5.

This means $d$ divided by 2 equals 7.5.
Undo dividing by 2 by multiplying by 2.
Work out 7.5 times 2 = 15, so $d = 15$ .
Check: 15 divided by 2 is 7.5.

Sometimes the letter is on the bottom of the fraction.

Definition

Denominator

The denominator is the bottom part of a fraction. In $30y\frac{30}{y}$ , the denominator is $y$ .

Example

Letter on the bottom

Solve $30y=6\frac{30}{y}=6$ .

Thirty divided into y equal groups gives groups of 6, so y is the number of equal groups.

Read it as “30 divided by $y$ equals 6.”
Ask: “30 divided by what gives 6?”
Work out 30 divided by 6 = 5, so $y = 5$ .
Check: 30 divided by 5 is 6.

Common Mistake

Letter on the bottom

For $30y=6\frac{30}{y}=6$ , the answer is not 30 times 6. Ask “30 divided by what is 6?” instead.

6. Checking your answer

Checking is one of the best ways to avoid losing easy marks.

Definition

Substitute

To substitute means to replace a variable with a number.

Example

Solve then check

Solve $5 + p = 13$ .

Addition can be in either order, so read this as $p + 5 = 13$ .
Undo add 5 by subtracting 5.
Work out 13 - 5 = 8, so $p = 8$ .
Substitute 8 back in: 5 + 8 = 13, so the answer works.

Exam technique

In the exam

Look for the operation attached to the letter: add, subtract, multiply or divide.
Use the inverse operation to get the letter on its own.
Check by putting your answer back into the original equation.

Self review

Check yourself

If $x + 7 = 20$ , what operation undoes add 7?
What does $6 a$ mean in words?
Why is $18 - y = 11$ different from $y - 18 = 11$ ?

Solving One Step Equations

What you'll learn

1. The equals sign means both sides match

When the missing number is being subtracted

2. Replacing the box with a letter

3. Undoing subtraction

4. Undoing multiplication

5. Undoing division and fractions

6. Checking your answer

Recap questions

You've reached the end

Solving One Step Equations

What you'll learn

1. The equals sign means both sides match

When the missing number is being subtracted

2. Replacing the box with a letter

3. Undoing subtraction

4. Undoing multiplication

5. Undoing division and fractions

6. Checking your answer

Recap questions

You've reached the end

Solving One Step Equations

What you'll learn

1. The equals sign means both sides match

When the missing number is being subtracted

2. Replacing the box with a letter

3. Undoing subtraction

4. Undoing multiplication

5. Undoing division and fractions

6. Checking your answer

Recap questions

You've reached the end

Solving One Step Equations Revision Guide

Solving One Step Equations

What you'll learn

1. The equals sign means both sides match

When the missing number is being subtracted

2. Replacing the box with a letter

3. Undoing subtraction

4. Undoing multiplication

5. Undoing division and fractions

6. Checking your answer

Recap questions

You've reached the end

Solving One Step Equations Revision Guide