Solving One Step Equations

What you'll learn

How missing-number boxes connect to algebra letters.
What an equation, an unknown, and a solution mean.
How to undo addition, subtraction, multiplication, and division.
How to check your answer quickly and avoid common traps.

1. Equations are balanced statements

An equation is like a balance: the left side and the right side have the same value.

Definition

Equation

An equation is a mathematical statement with an equals sign, such as $x + 6 = 18$ . The equals sign means “has the same value as”.

In algebra, a letter such as $x$ , $m$ , or $a$ often stands for a number we do not know yet. This is called the unknown or variable. To solve an equation means to find the value of the unknown that makes the equation true.

A balance picture is useful because it reminds you that both sides of an equation must stay equal.

A balance scale showing x plus 6 blocks equal to 18 blocks, then subtracting 6 from both sides

Key Idea

Keep the equation balanced

If you change one side of an equation, you must make the matching change to the other side. For one-step equations, this usually means using one inverse operation.

Worked example: missing boxes

Example

Finding missing numbers

For $\boxed{\phantom{0}} = 19$ , ask: “What number must be added to 7 to make 19?”
Use the related subtraction fact.
$19 - 7 = 12$
So the missing number in $\boxed{\phantom{0}} = 19$ is 12.
For $\boxed{\phantom{0}} = 11$ , ask: “What number was taken away from 18 to leave 11?”
Use the difference between 18 and 11.
$18 - 11 = 7$
So the missing number in $\boxed{\phantom{0}} = 11$ is 7.

2. From boxes to letters

A letter in an equation is just like a missing-number box. For example, $x + 5 = 16$ means:

“What number, when 5 is added, gives 16?”

Definition

One-step equation

A one-step equation is an equation that can be solved by undoing just one operation, such as adding, subtracting, multiplying, or dividing.

Worked example: a letter instead of a box

Example

Solving an addition equation

Start with the equation.
$x + 5 = 16$
The unknown has 5 added to it, so undo this by subtracting 5.
$x+5=16x=16−5x=11\begin{aligned} x + 5 &= 16 \\ x &= 16 - 5 \\ x &= 11 \end{aligned}$
The solution is $x = 11$ .
Check by putting 11 back into the original equation: 11 + 5 = 16, so it works.

Tip

Think of the question in words

For $x + 5 = 16$ , you can read it as: “What number plus 5 equals 16?” This often makes Grade 2 algebra feel much less mysterious.

3. Inverse operations for adding and subtracting

Definition

Inverse operations

Inverse operations are operations that undo each other. Addition and subtraction are inverse operations. Multiplication and division are inverse operations.

For example:

To undo adding 9, subtract 9.
To undo subtracting 4, add 4.

Worked example: subtracting from the unknown

Example

Solving a subtraction equation

Start with the equation.
$m - 6 = 13$
The unknown has 6 subtracted from it, so undo this by adding 6.
$m−6=13m=13+6m=19\begin{aligned} m - 6 &= 13 \\ m &= 13 + 6 \\ m &= 19 \end{aligned}$
The solution is $m = 19$ .
Check: 19 minus 6 gives 13, so the answer is correct.

When the unknown comes after a minus sign

Equations like $20 - p = 12$ need extra care. Here, $p$ is not the starting number. It is the amount being taken away.

Common Mistake

Treating every subtraction equation the same way

In $20 - p = 12$ , do not write $p = 20 + 12$ . The unknown is the number being subtracted, so you need the difference between 20 and 12.

Example

Solving when the letter is being subtracted

Start with the equation.
$20 - p = 12$
Ask: “What must be taken away from 20 to leave 12?”
Find the difference between 20 and 12.
$20 - 12 = 8$
So $p = 8$ .
Check: 20 minus 8 gives 12.

4. Multiplication equations

When a number is written next to a letter, it means multiplication. For example, $6 a$ means $\times a$ .

Definition

Coefficient

A coefficient is the number multiplying a variable. In $6 a$ , the coefficient of $a$ is 6.

To solve a multiplication equation, divide by the coefficient.

Worked example: solving by dividing

Example

Solving a multiplication equation

Start with the equation.
$8 g = 48$
This means 8 lots of $g$ make 48, so divide by 8.
$8g=48g=48÷8g=6\begin{aligned} 8g &= 48 \\ g &= 48 \div 8 \\ g &= 6 \end{aligned}$
The solution is $g = 6$ .
Check: 8 lots of 6 make 48.

5. Division equations

A fraction line can mean division. For example, $d2=6.5\frac{d}{2} = 6.5$ means “ $d$ divided by 2 equals 6.5”.

To undo division, multiply.

Worked example: solving a division equation

Example

Solving when the unknown is divided

Start with the equation.
$f3=7\frac{f}{3} = 7$
The unknown has been divided by 3, so undo this by multiplying by 3.
$f3=7f=7×3f=21\begin{aligned} \frac{f}{3} &= 7 \\ f &= 7 \times 3 \\ f &= 21 \end{aligned}$
The solution is $f = 21$ .
Check: 21 divided by 3 gives 7.

Division with decimals

Example

Solving with a decimal answer

Start with the equation.
$d2=6.5\frac{d}{2} = 6.5$
Undo dividing by 2 by multiplying by 2.
$d2=6.5d=6.5×2d=13\begin{aligned} \frac{d}{2} &= 6.5 \\ d &= 6.5 \times 2 \\ d &= 13 \end{aligned}$
The solution is $d = 13$ .

6. Repeated letters

Sometimes the same letter is added more than once, such as $h + h + h = 24$ .

Because each $h$ has the same value, $h + h + h$ means 3 lots of $h$ , so it can be written as $3 h$ .

Worked example: repeated unknowns

Example

Turning repeated letters into multiplication

Start with the equation.
$n + n + n = 45$
Rewrite the left side as 3 lots of $n$ .
$3 n = 45$
Divide by 3 to find one $n$ .
$3n=45n=45÷3n=15\begin{aligned} 3n &= 45 \\ n &= 45 \div 3 \\ n &= 15 \end{aligned}$
The solution is $n = 15$ .

7. When the letter is underneath

In equations like $36y=9\frac{36}{y} = 9$ , the unknown is the number you are dividing by. This number is called the divisor.

Read it carefully:

“36 divided by what number gives 9?”

Common Mistake

Dividing by zero

A divisor cannot be zero, because division by zero is not allowed.

Worked example: unknown as the divisor

Example

Solving when the letter is the divisor

Start with the equation.
$48y=8\frac{48}{y} = 8$
Read it as: “48 divided by what number gives 8?”
Use the related multiplication fact: 8 times what number gives 48?
$\times 6 = 48$
So $y = 6$ .
Check: 48 divided by 6 gives 8.

8. Checking your answer

To check an answer, substitute it back into the original equation. Substitute means replace the letter with the number you found.

Example

Checking by substitution

Suppose you solve $5 + p = 14$ and get $p = 9$ .
Substitute 9 for $p$ in the original equation.
$5 + 9 = 14$
Both sides are equal, so $p = 9$ is correct.

Exam technique

In the exam

Identify what operation is happening to the unknown: adding, subtracting, multiplying, or dividing.
Use the inverse operation and show one clear line of working, even for a 1-mark question.
Check tricky cases carefully, especially equations like $20 - m = 12$ or $36y=9\frac{36}{y} = 9$ .

Self review

Check yourself

If $x + 7 = 19$ , what operation undoes adding 7?
In $4 a = 28$ , what does the 4 mean?
In $25 - m = 18$ , is $m$ the starting number, the amount taken away, or the answer?

Solving One Step Equations

What you'll learn

How missing-number boxes connect to algebra letters.
What an equation, an unknown, and a solution mean.
How to undo addition, subtraction, multiplication, and division.
How to check your answer quickly and avoid common traps.

1. Equations are balanced statements

An equation is like a balance: the left side and the right side have the same value.

Definition

Equation

An equation is a mathematical statement with an equals sign, such as $x + 6 = 18$ . The equals sign means “has the same value as”.

A balance picture is useful because it reminds you that both sides of an equation must stay equal.

A balance scale showing x plus 6 blocks equal to 18 blocks, then subtracting 6 from both sides

Key Idea

Keep the equation balanced

If you change one side of an equation, you must make the matching change to the other side. For one-step equations, this usually means using one inverse operation.

Worked example: missing boxes

Example

Finding missing numbers

For $\boxed{\phantom{0}} = 19$ , ask: “What number must be added to 7 to make 19?”
Use the related subtraction fact.
$19 - 7 = 12$
So the missing number in $\boxed{\phantom{0}} = 19$ is 12.
For $\boxed{\phantom{0}} = 11$ , ask: “What number was taken away from 18 to leave 11?”
Use the difference between 18 and 11.
$18 - 11 = 7$
So the missing number in $\boxed{\phantom{0}} = 11$ is 7.

2. From boxes to letters

A letter in an equation is just like a missing-number box. For example, $x + 5 = 16$ means:

“What number, when 5 is added, gives 16?”

Definition

One-step equation

A one-step equation is an equation that can be solved by undoing just one operation, such as adding, subtracting, multiplying, or dividing.

Worked example: a letter instead of a box

Example

Solving an addition equation

Start with the equation.
$x + 5 = 16$
The unknown has 5 added to it, so undo this by subtracting 5.
$x+5=16x=16−5x=11\begin{aligned} x + 5 &= 16 \\ x &= 16 - 5 \\ x &= 11 \end{aligned}$
The solution is $x = 11$ .
Check by putting 11 back into the original equation: 11 + 5 = 16, so it works.

Tip

Think of the question in words

For $x + 5 = 16$ , you can read it as: “What number plus 5 equals 16?” This often makes Grade 2 algebra feel much less mysterious.

3. Inverse operations for adding and subtracting

Definition

Inverse operations

Inverse operations are operations that undo each other. Addition and subtraction are inverse operations. Multiplication and division are inverse operations.

For example:

To undo adding 9, subtract 9.
To undo subtracting 4, add 4.

Worked example: subtracting from the unknown

Example

Solving a subtraction equation

Start with the equation.
$m - 6 = 13$
The unknown has 6 subtracted from it, so undo this by adding 6.
$m−6=13m=13+6m=19\begin{aligned} m - 6 &= 13 \\ m &= 13 + 6 \\ m &= 19 \end{aligned}$
The solution is $m = 19$ .
Check: 19 minus 6 gives 13, so the answer is correct.

When the unknown comes after a minus sign

Equations like $20 - p = 12$ need extra care. Here, $p$ is not the starting number. It is the amount being taken away.

Common Mistake

Treating every subtraction equation the same way

In $20 - p = 12$ , do not write $p = 20 + 12$ . The unknown is the number being subtracted, so you need the difference between 20 and 12.

Example

Solving when the letter is being subtracted

Start with the equation.
$20 - p = 12$
Ask: “What must be taken away from 20 to leave 12?”
Find the difference between 20 and 12.
$20 - 12 = 8$
So $p = 8$ .
Check: 20 minus 8 gives 12.

4. Multiplication equations

When a number is written next to a letter, it means multiplication. For example, $6 a$ means $\times a$ .

Definition

Coefficient

A coefficient is the number multiplying a variable. In $6 a$ , the coefficient of $a$ is 6.

To solve a multiplication equation, divide by the coefficient.

Worked example: solving by dividing

Example

Solving a multiplication equation

Start with the equation.
$8 g = 48$
This means 8 lots of $g$ make 48, so divide by 8.
$8g=48g=48÷8g=6\begin{aligned} 8g &= 48 \\ g &= 48 \div 8 \\ g &= 6 \end{aligned}$
The solution is $g = 6$ .
Check: 8 lots of 6 make 48.

5. Division equations

A fraction line can mean division. For example, $d2=6.5\frac{d}{2} = 6.5$ means “ $d$ divided by 2 equals 6.5”.

To undo division, multiply.

Worked example: solving a division equation

Example

Solving when the unknown is divided

Start with the equation.
$f3=7\frac{f}{3} = 7$
The unknown has been divided by 3, so undo this by multiplying by 3.
$f3=7f=7×3f=21\begin{aligned} \frac{f}{3} &= 7 \\ f &= 7 \times 3 \\ f &= 21 \end{aligned}$
The solution is $f = 21$ .
Check: 21 divided by 3 gives 7.

Division with decimals

Example

Solving with a decimal answer

Start with the equation.
$d2=6.5\frac{d}{2} = 6.5$
Undo dividing by 2 by multiplying by 2.
$d2=6.5d=6.5×2d=13\begin{aligned} \frac{d}{2} &= 6.5 \\ d &= 6.5 \times 2 \\ d &= 13 \end{aligned}$
The solution is $d = 13$ .

6. Repeated letters

Sometimes the same letter is added more than once, such as $h + h + h = 24$ .

Because each $h$ has the same value, $h + h + h$ means 3 lots of $h$ , so it can be written as $3 h$ .

Worked example: repeated unknowns

Example

Turning repeated letters into multiplication

Start with the equation.
$n + n + n = 45$
Rewrite the left side as 3 lots of $n$ .
$3 n = 45$
Divide by 3 to find one $n$ .
$3n=45n=45÷3n=15\begin{aligned} 3n &= 45 \\ n &= 45 \div 3 \\ n &= 15 \end{aligned}$
The solution is $n = 15$ .

7. When the letter is underneath

In equations like $36y=9\frac{36}{y} = 9$ , the unknown is the number you are dividing by. This number is called the divisor.

Read it carefully:

“36 divided by what number gives 9?”

Common Mistake

Dividing by zero

A divisor cannot be zero, because division by zero is not allowed.

Worked example: unknown as the divisor

Example

Solving when the letter is the divisor

Start with the equation.
$48y=8\frac{48}{y} = 8$
Read it as: “48 divided by what number gives 8?”
Use the related multiplication fact: 8 times what number gives 48?
$\times 6 = 48$
So $y = 6$ .
Check: 48 divided by 6 gives 8.

8. Checking your answer

To check an answer, substitute it back into the original equation. Substitute means replace the letter with the number you found.

Example

Checking by substitution

Suppose you solve $5 + p = 14$ and get $p = 9$ .
Substitute 9 for $p$ in the original equation.
$5 + 9 = 14$
Both sides are equal, so $p = 9$ is correct.

Exam technique

In the exam

Identify what operation is happening to the unknown: adding, subtracting, multiplying, or dividing.
Use the inverse operation and show one clear line of working, even for a 1-mark question.
Check tricky cases carefully, especially equations like $20 - m = 12$ or $36y=9\frac{36}{y} = 9$ .

Self review

Check yourself

If $x + 7 = 19$ , what operation undoes adding 7?
In $4 a = 28$ , what does the 4 mean?
In $25 - m = 18$ , is $m$ the starting number, the amount taken away, or the answer?

Solving One Step Equations

What you'll learn

1. Equations are balanced statements

Worked example: missing boxes

2. From boxes to letters

Worked example: a letter instead of a box

3. Inverse operations for adding and subtracting

Worked example: subtracting from the unknown

When the unknown comes after a minus sign

4. Multiplication equations

Worked example: solving by dividing

5. Division equations

Worked example: solving a division equation

Division with decimals

6. Repeated letters

Worked example: repeated unknowns

7. When the letter is underneath

Worked example: unknown as the divisor

8. 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. Equations are balanced statements

Worked example: missing boxes

2. From boxes to letters

Worked example: a letter instead of a box

3. Inverse operations for adding and subtracting

Worked example: subtracting from the unknown

When the unknown comes after a minus sign

4. Multiplication equations

Worked example: solving by dividing

5. Division equations

Worked example: solving a division equation

Division with decimals

6. Repeated letters

Worked example: repeated unknowns

7. When the letter is underneath

Worked example: unknown as the divisor

8. Checking your answer

Recap questions

You've reached the end

Solving One Step Equations Revision Guide