# Introduction

The ability to simplify and manage algebraic equations is a fundamental tool you will need in order to successfully complete your studies. You will have covered many of what is going to be described below in earlier courses. However it will serve as a useful reminder tool when it comes to working on equations in later modules.

Some of you may already know this and may decide to skip sections. However it will serve as a tool for finding out where certain parts of any examples have gone if more than 1 step is combined.

## Expressions and Terms

All algebraic equations can be considered as a series of expressions. On each side of the equals '=' sign there will be an expression. For example:

```$5x + 2x = 7$

contains the expressions $(5x + 2x)$ and $(7)$
```

Within each expression, there will be a number of terms. A term is any variable, number, or algebraic letter. In the equation above we can say that there are three terms. These three terms are:

```$5x$ $2x$ and $7$
```

Now what is key to all equations is simplifying what they say, that is reducing the length of the expression by collecting like terms together. We have seen that the equation above technically has three terms. Let us break down a term further.

```The term $5x$ has a coefficient of $5$ and a variable of $x$
```

When you simplify an expression, you simply collect the like terms together. In other words, you collect the terms with the same variables together.

So taking our very first equation: $5x + 2x = 7$ we can see that the terms $5x$ and $+2x$ contain the same variable $x$. They can therefore be added together to obtain $7x$

So, From above we can see that like terms can only be collected together in order to simplify an expression and hence an equation. Terms which do not contain identical variables cannot be simplified. The exaples below show collecting like and unlike terms together.

### Example 1

Simplify the expression $5x^2 + 2x + 3x + 5 + 7$

The Table to the right shows the number of terms with each variable.

Variable Number of Terms
$x^2$ 1
$x$ 2
No Variable 2

Reading this table we can see that:

• There is only 1 term in $x^2$ ($5x^2)$ and so it cannot be simplified.
• There are two terms in $x$ ($+2x$ and $+3x$) so these can be added.
• There are two terms with no variable $+5$ and $+7$ and these can be added.

Therefore after simplifying the expression we finish with $5x^2 + 5x + 12$

### Example 2

Simplify the expression $3x^2 + 2x + 2y - 4y - 3x +x^2 -xy$

Using the rules stated above the expression can be simplified to:

$4x^2 - x - 2y - xy$

### Single Variables

In example 2 above we ended up with the term $-x$. It is important to note that when there is just a variable, with no apparent coefficient, the coefficient is in fact 1.

• $-x$ is the same as $-1x$
• $xy$ is the same as $1xy$