Problems are often answered in mathematics by solving equations.

To solve them, we need to find what number the letter in the equation represents. That number is called the solution of the equation.

There are usually several ways to solve an equation. If the method you choose to use always gives you the correct answer, then keep using this method!

We are going to use the method:

**Change side, change operation**

- Question
Solve the equation \(x + 5 = 12\)

Add becomes subtract.

- Question
Solve the equation \(3x - 15 = 9\)

Subtract becomes add and multiply becomes divide.

When we have letters on both sides of the equation, we need to use the rule:

**Letters to the left, numbers to the right**

- Question
Solve the equation \(7x - 3 = 3x + 17\)

\[7x - 3 = 3x + 17\]

\[7x - 3x = 17 + 3\]

\[4x = 20\]

\[x = 20 \div 4\]

\[x = 5\]

- Question
Solve the equation \(5p - 2 = 7p + 12\)

\[5p - 2 = 7p + 12\]

\[5p - 7p = 12 + 2\]

\[- 2p = 14\]

\[p = 14 \div - 2\]

\[p = - 7\]

An equation may be used to model a situation. For example:

- Question
Mr and Mrs Wallace hire a van for moving house. They see this advert in the local paper and decide to use this company.

When they hand the van back, their bill comes to £59. How many hours did they hire the van for?

First we need to put this information into an equation.

\(x\) is the number of hours they hired the van for.This is what we have to find.

Their total bill will be 17 plus 6 times \(x\).

We usually write this as \(6x + 17\).

\[6x + 17 = 59\]

\[6x = 59 - 17\]

\[6x = 42\]

\[x = 42 \div 6\]

\[x = 7\]

Therefore they hired the van for 7 hours.