# Using equations to solve problems

**Constructing an equation**

When constructing an equation, the number *n* can be used to represent the solution of the equation.

**Solving the equation**

An example of solving an equation to a problem is as follows:

= 2(*n *– 9) =

To solve this equation, the following steps are required:

= 10(*n *– 9) = *n* (multiply both sides by 5)

= 10*n *– 90 = *n *(expand the brackets)

= 10*n* = *n +90 *(add 90 to both sides)

= 9*n* =* +90 *(subtract *n* from both sides)

= *n* = 10

**Using equations to solve problems**

A worked example is as follows, the sum of the ages of John and Michael is 66 years. In two year’s time John will be three times Michael’s age.

How old is Michael now?

Michael’s age will be represented as *a*.

The first piece of information will indicate that John’s current age as 66 – *a.*

The second piece of information will indicate that John’s current age as 3(*a* + 2) – 1

This can then be simplified to,

= 3(*a* + 2) – 1 = 3*a* + 6 – 1

= 3*a *+ 5

The two expressions can then be written as an equation to find Michael’s age:

= 66 – *a *= 3*a *+ 5 (subtract 5 from both sides)

= 61 – *a *= 3*a* (add *a* from both sides)

= 61* *= 2*a* (divide both sides by 2)

= *a = *30.5