Solving+Linear+Equations

What is an equation?
Equations are made up of two expressions on either side of an equals sign, like x + 2 = 1 To solve an equation, you need to find the values of the missing numbers.

Example
'I think of a number, add four, and the answer is seven.' Written algebraically, this statement is '**x + 4 = 7**', where 'x' represents the number you thought of. 'x + 4 = 7' is an example of an **algebraic equation**. 'x' represents an unknown number. The number you first thought of must be three (3 + 4 = 7). Therefore, x = 3 is the solution to the equation x + 4 = 7.

Balancing an Equation ** An equation is like a weighing scale - both sides should always be perfectly balanced. To solve the equation you need to find the value of missing numbers and perform the same operation to each side.

Example 1
For example, suppose you are trying to find out how many sweets are in the bag shown here. By subtracting three sweets from each side, the scales remain balanced.

You can now see that one bag is equivalent to two sweets. Written algebraically, this is: x + 3 = 5 Subtract 3 from both sides, to give: x = 2

Example 2
In this case, two bags of sweets are equivalent to six sweets:

To find the equivalent of one bag, divide both sides in half:

Written algebraically, this is: 2x = 6 Divide both sides by 2, to give: x = 3

Questions on solving equations

 * 1) 3a = 9
 * 2) 5a = 15
 * 3) 4a = 24

One-step equation game


 * Now lets look at some more complex equations **

Sometimes an equation will have multiples of an unknown, eg, 5y = 20. To solve this you need to get the unknown on its own. To do this, divide both sides by 5. 5y = 20 5y ÷ 5 = 20 ÷ 5 y = 4 Sometimes an equation will have multiples of an unkown and other numbers, eg, 3x + 2 = 8 In equations of this type, your aim is to get all the 'x's (or unknowns) on one side and all the numbers on the other.

Example
Let's solve the equation 3x + 2 = 8. We can show this in a picture like this, where each bag is an 'x'.


 * [[image:http://www.bbc.co.uk/schools/ks3bitesize/maths/images/equations4.gif width="516" height="191" caption="Three bags of sweets plus two sweets and eight sweets blancing on a scale"]] ||
 * Three bags of sweets plus two sweets and eight sweets blancing on a scale ||

3x + 2 = 8 We want to get the 'x' on it's own. Start by subtracting 2 from both sides:


 * [[image:http://www.bbc.co.uk/schools/ks3bitesize/maths/images/equations5.gif width="516" height="191" caption="Three bags of sweets and six sweets balancing on a scale"]] ||
 * Three bags of sweets and six sweets balancing on a scale ||

3x + 2 - 2 = 8 - 2 3x = 6 Then divide by 3: So x = 2


 * [[image:http://www.bbc.co.uk/schools/ks3bitesize/maths/images/equations8.gif width="516" height="191" caption="A bag of sweets and two sweets balancing on a scale"]] ||
 * A bag of sweets and two sweets balancing on a scale ||

Questions on more complex equations

 * 1) 2x + 4 - 8 = 4
 * 2) 5x ÷ 3 = 5
 * 3) (5x + 10) ÷ 2 = 20

Now that you've learnt how to solve these equations why not have a go at this activity:

Introduction to equations