## Algebraic Equations

### Introduction

The fundamental idea on which algebraic equation are based is that an equation is exactly like a set of balance scales. If you do exactly the same thing to both sides of the equation then the equation will always balance. If you add the same amount to both sides or subtract the same amount to both sides the equation will balance. If you multiply the Left Hand Side (LHS) by the same amount as the Right Hand Side (RHS), then the equation will still balance, and so on. This is all analogous with a set of balance scales. With an algebraic equation we can go further and do exotic things and, for example, take the square root of both side and the equation will still balance.

Once you have this fundamental idea in your mind, we can go on to talk about how to use these allowed opersations systematically to solve an equation.

Another thing in passing - since we commonly use the letter 'x' in algebra, the use of the 'X' sign for multiplication is strongly discouraged.

### Basic Equations - No 1

Consider

x + 5 = 8

Although you might be able to guess what the answer is, I would to explain the algebraic method. The algebraic method would be (as always) to do exactly the same thing to both LHS and RHS.

Here I will subtract 5 from both sides

x = 3

As a second example, consider

x -4 = 6

The algebraic method would be to do exactly the same thing to both LHS and RHS.

Here I will add 4 to both sides

x = 10

Try these examples before proceeding

 Quick Quiz a)   x + 3 = 8   b)  x + 4 = 7   c)   x + 9 = 3 d)  x - 11 = 15   e)   x - 23 = -51

### Basic Equations - No 2

Consider

2x = 8

Again, although you might be able to guess what the answer is, I would to explain the algebraic method. The algebraic method would be (as always) to do exactly the same thing to both LHS and RHS.

Here I will divide both sides by 2

x = 4

As a second example, consider

$\frac{x}{6} = 6$

The algebraic method would be to do exactly the same thing to both LHS and RHS.

Here I will multiply both sides by 6

x = 36

Try these examples before proceeding

 Quick Quiz a) $4x = 8$ b)$\frac{x}{3} = 5$ c) $3x = 12$ d)$\frac{x}{4} = 2$ e) $12x = 48$

### Basic Equations - No 3

If I can now make use of the methods used in the last two sections simultaneously in order to show how to tackle a more 'complicated' expression in a systematic way.

Consider

3x - 7 = x - 5

Firstly, I am going to use the methods of addition or subtraction to get every term containing an x on to the same side, and everything which is just a number on to the other side.

Secondly, I am going to use the operations of multiplication or division to obtain x

The first process involves two steps

Subtract x from both sides

2x - 7 = - 5

2x = 2

The second process involves just one step. We are told what 2x equals but I actually want to know what x equals, so

Divide both sides by 2

x = 1

As a second example, consider

6x + 9 = 25 - 2x

The first process involves two steps

8x + 9 = 25

Subtract 9 from both sides

8x = 16

The second process involves just one step. We are told what 8x equals but I actually want to know what x equals, so

Divide both sides by 8

x = 2