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Solve equations without guessing.

Learn two calm ways to solve equations: the inverse method and the balance method. Both are built around one idea: undo what is happening to the unknown.

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Written lesson · Worked examples · Practice questions

Year 7 Mathematics

Algebra

Solving Equations: Inverse and Balance Methods

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Understand equations as balanced statements.

Use inverse operations to undo steps.

Use the balance method to keep both sides equal.

Learn

What an equation is really saying

An equation is a sentence with an equals sign. It says the left side and the right side have the same value.

Solving an equation means finding the value of the unknown that makes the sentence true. We are not guessing. We are carefully undoing the operations around the unknown.

There are two useful ways to think about this: the inverse method and the balance method. They look different, but they are really the same idea wearing two different jackets.

Method 1: The inverse method

Inverse means opposite. Addition and subtraction undo each other. Multiplication and division undo each other.

If the equation says x + 3, the unknown has had 3 added to it. To get back to x, subtract 3.

\begin{aligned}x+3&=11\\x&=11-3\\x&=8\end{aligned}

I know to subtract 3 because +3 is the operation attached to x, and subtraction is the inverse operation.

Method 2: The balance method

The balance method treats the equals sign like a balanced scale. Whatever you do to one side, you must do to the other side.

For x + 3 = 11, subtract 3 from both sides. The left side becomes x, and the right side becomes 8.

\begin{aligned}x+3&=11\\x+3-3&=11-3\\x&=8\end{aligned}

This works because both sides changed by the same amount, so the equation stayed balanced.

Worked examples

Three common equation types

Example 1: Addition around x

x+6=15

The x has 6 added to it. Undo +6 with -6.

\begin{aligned}x+6-6&=15-6\\x&=9\end{aligned}

Example 2: Subtraction around x

x-4=10

The x has 4 subtracted from it. Undo -4 with +4.

\begin{aligned}x-4+4&=10+4\\x&=14\end{aligned}

Example 3: Multiplication around x

5x=35

5x means 5 multiplied by x. Undo multiplication by 5 with division by 5.

\begin{aligned}\frac{5x}{5}&=\frac{35}{5}\\x&=7\end{aligned}

Practice

Try a few

For each one, first ask: what is happening to x? Then undo it.

Question 1

Solve

x+5=12

Hint: The +5 is attached to x, so undo it with -5.

Show answer

x=7

Question 2

Solve

3x=18

Hint: The 3 is multiplying x, so undo it by dividing by 3.

Show answer

x=6

Question 3

Solve

x-4=9

Hint: The -4 is attached to x, so undo it with +4.

Show answer

x=13

Common traps

What students often mix up

Trap: Moving a number to the other side without knowing why. Fix: name the inverse operation.
Trap: Changing only one side. Fix: if you use the balance method, do the same operation to both sides.
Trap: Thinking 4x means x + 4. Fix: 4x means 4 multiplied by x.

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