## What is Algebra?

Algebra is simply the use of letters, called pronumerals, to represent unknown numbers. We use pronumerals in a question or equation when we don’t know the numerical value of something. Common letters used for the pronumerals are ‘x’, ‘y’, ‘z’ and ‘a’, but any letter can be used.

Like terms can be added or subtracted. For example 2y and 3y are **like** terms. Therefore:

2y + 3y = 5y

Unlike terms can’t be added or subtracted. For example 2y and 3y² are **unlike** terms. Therefore:

2y + 3y² = 2y + 3y²

Remember your directed numbers as shown below

#### Like signs make “+”

**Example 1a:**

= +3 + +3

= +3+3

= +6

#### Unlike signs make “-“

**Example 1c:**

= +3 + -3

= +3 – 3

= 0

**Example 1b:**

= +3 – -3

= +3+3

= +6

**Example 1d:**

= +3 – +3

= +3 – 3

= 0

## Multiplication of Terms

The terms do not need to be “like” in order to multiply. See the examples below:

#### Like signs make “+”

**Example 2a:**

= +3b × +3b

= +9b²

#### Unlike signs make “-“

**Example 2c:**

= -3b × +3b

= -9b²

**Example 2b:**

= -3b × -3b

= +3 × +3

= +9b²

**Example 2d:**

= +3b × -3b

= +3 × – 3

= -9b²

## Expansion of Brackets

When expanding the brackets, each term in the bracket is multiplied by the expression outside the bracket.

**Example 3a:**

5(a+2b)

= 5 × a + 5 × 2b

= 5a + 10b

**Example 3b:**

-2(-1 + 3b)

= -2 × -1 + -2 × 3b

= 2 – 6b

## Factorising

Factorising is essentially the reverse of expanding the brackets. Factorising involves finding the highest common factor in an expression. For example:

= 5y + 10 (common factor is 5)

In the example above, 5 is the common factor because 5 can be taken out of 5y, to leave “y”, and 5 can be taken out of 10, to leave 2.

Take the common factor out and place the remainder of each term in a bracket:

**Example 4a:**

= 5y + 10

= 5(y+2)

**Example 4b:**

= 2ba + 10bd (common factor is 2b)

= 2b(a + 5d)

## Cancelling

Normal fractions can be simplified by cancelling. An example of this is shown below:

Algebraic fractions can also be simplified. Cancelling numbers or pronumerals that divide evenly on the top and bottom lines of the fraction. This is shown below:

## Equations

Equations allow us to reduce complex terms to simple terms. We can then utilise these equations to solve many different problems.

Imagine a pair of scales, whereby the left-hand side equals the right-hand side.

The main rule is to keep the equation balanced

If we double one side, to keep the balance, the other side must be doubled as well.

Therefore, as long as we do the same to both sides, the equation will remain balanced

By working backwards, complex equations can be reduced to simple terms.

5x = 10

÷5 ÷5

∴ x=2

+ and – are inverse operations

× and ÷ are inverse operations