How can algebraic expressions be simplified?

In algebra, long expressions can be simplified which makes them easier to solve.

We can simplify alebraic expressions by collecting together "like terms".

When looking at an expressions such as:
5x + 4x - 2 - 2x + 10
the terms with the same letter can be collected.

For example:
5x + 4x - 2x in the equation above all have the same letter and can be added and substracted. This equals 7x (5 + 4 - 2)

The numbers both do not have any letter after them so also can be collected together as they are "like terms".

10 - 2 = 8

So 5x + 4x - 2 - 2x + 10 simplified is 7x + 8


Sometimes expressions will have several different terms or letters in them.

For example: 

5x + 4y - 3x + 4y - 7z

The same method is used with all the x terms being collected together, then all the y terms and then all the z terms.

5x - 3x = 2x

4y + 4y = 8y

- 7z = - 7z

So 5x + 4y - 3x + 4y - 7z simplified is 2x + 8y - 7z


Some questions might ask you to multiply out brackets

It is important to remember that:
2a means "2 times a" 
ab means "a times b"
 means "a times a"

So for example in the expression:
5 ( 3x + 5 )

the number on the outside of the brackets is multiplied by all the terms inside the brackets in multiple steps.

FIRST: 5 times 3x = 15x
SECOND: 5 times 5 = 25

therefore: 5 ( 3x + 5 ) = 15x + 25

Be careful in some expressions. 
If the sign inside the expression is negative for example:

3 ( 2x - 2 )
Then remember that the sign is also multiplied.
3 times - 2 = - 6

Similarly if the term outside the brackets is a letter such as:
a ( a + 5 )

then it is important to remember a times a is a2



Some expressions can have two brackets. In this case everything in the first bracket needs to be multiplied by everything in the second bracket.

An easy way of doing this is to take each term in turn and multiply it by each term in the second bracket. 
For example:

( x + 4 )( x + 3 )

the x in the first bracket is taken first

FIRST: x is multiplied by the the x in the second bracket = x2
SECOND: x is multiplied by the 3 in the second bracket = 3x

then the 4 in the first bracket is done the same

FIRST: 4 is multiplied by x in the second bracket = 4x
SECOND: 4 is multiplied by 3 in the second bracket = 12

all these terms are then collected together:
x2 + 3x + 4x + 12

this is then simplified to x2 + 7x + 12


An easier way of remembering how to expand brackets, is to use the acronym F.O.I.L

F = first 
O = outer
I = inner
L = last

this shows the order which you should multiply out the terms.

( x + 3 )( x + 4 )
First = x times x = x2
Outer = x times 4 = 4x

Inner = 3 times x = 3x
Last = 3 times 4 = 12


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