 Problem solving
Lesson

Now comes the really exciting part where you can use all the rules and principles of algebra that you have learnt to solve problems!

Algebra is all about describing patterns and relationships using symbols. As you have seen before, these symbols are usually letters such as $x$x or $a$a and they describe an unknown quantity of something.

In this chapter you will need to think about how you can write a worded problem as an algebraic expression, or how you can describe a visual pattern using an algebraic expression.

#### Examples

The sum of three consecutive numbers is $99$99.

a) If the first number is $x$x, write an expression for the sum of of the three numbers, expressing your answer in its simplest form.

Think: Consecutive numbers are integers that come one after the other. So if the first number is $x$x, one number more than this can be written as ($x+1$x+1) and the number after this would be ($x+1+1$x+1+1) which we can simplify to ($x+2$x+2).

So the sum of these three numbers can be expressed as:

$x+x+1+x+2$x+x+1+x+2 = $99$99

This can be simplified to:

$3x+3$3x+3 = $99$99

b) Hence or otherwise, find the value of $x$x.

Think: We can solve this equation:

$3x+3$3x+3 = $99$99 (Subtract 3)

$3x$3x = $96$96 (Divide by 3)

$x$x = $32$32

c) Hence, what are the three numbers?

Think: We've just worked out that the first number, $x$x, is 32.

Do: The three numbers are $32$32, $33$33 and $34$34.

Check: $32+33+34=99$32+33+34=99

#### More Examples

##### Question 1

A $23$23cm wire is cut into $2$2 pieces such that the length of the smaller piece is $4$4 cm. The longer piece is then bent to form a rectangle of width $3$3 cm.

1. What is the length of the longer piece?

2. Hence, what is the length of the rectangle?

##### Question 2

Each week, Aaron earns $\$m$$m more than Paul. Each week, Uther earns \u$$u more than Aaron.

If Paul earns $\$440440 per week, how much does Uther earn per week?

##### Question 3

The local stationery shop has a sale on boxes of pens. Each box contains an unknown number of pens.

Hermione bought $13$13 boxes of pens.

Caitlin bought $12$12 boxes of pens.

1. Write an expression for the total number of pens the two girls bought.

Use the variable $p$p to represent the number of pens in each box.

### Outcomes

#### 9D.NA2.09

Solve problems that can be modelled with first-degree equations, and compare algebraic methods to other solution methods