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What is Simplification

The process of reducing the algebraic expression or equation to its simplest form. The resulting expression or equation should have no brackets no multiple occurrences of the same term.

The above is a type of dictionary definition of simplification, for the moment lets forget about brackets. What do we mean by multiple occurrences of the same term, what is a term. Here is a simple equation:

3x + 2y - 2x + 4xy + 5 - yx = 9

What in this equation are terms, in fact that is simple if we put a comma in front of any plus signs, minus signs and replace the equal sign with a comma and remove all the spaces now write down the result:
3x,+2y,-2x,+4xy,+5,-yx,9

what we have now is a list of the terms in the equation, we can also leave the plus signs of as a plus sign is assumed if not shown but we must leave the minus sign So now our list of terms looks like this:
3x, 2y, -2x, 4xy, 5, -yx, 9

Are any terms in this list the same, there are no identical terms but there are some terms that we regard as the same because when we say the same we do not mean identical we in fact mean of the same type. Two terms of the same type are 3x and -2x, these are of the same type because they are both a number followed by the unknown x, these terms are different from the term 2y because this is a different unknown. We can combine terms of the same type simply by adding the number part of the term or subtracting when a terms number part has a minus sign. If there is no number part then the number of that term is one or if it just has a minus sign its number part is -1. Combining the two terms 3x and -2x thus gives us the single term 1x but we write this as x by itself.

Another pair of terms that are of the same type and can be combined are 5 and 9. Here there are no unknowns so in effect they are still just numbers and can be treated as such but note these two numbers are on opposite sides of the equal sign (=), because of this we cannot just add the two number terms. We have to first get both number terms on to the same side of the equation, this is in fact quite simple you just move the term you want to the other side of the equal sign but when you do this the sign of the term is reversed. So we could move the 5 to the right of the equal sign and making it -5 or we could move the 9 to the left side of the equal sign and make it -9. the resulting equation would be:

x + 2y + 4xy - yx = 9 - 5

OR
x + 2y + 4xy + 5 - yx - 9 = 0

Now the number terms can be combined.

The reason why the sign of a term is changed when it is moved to the other side of the equal sign is as follows. An equation is like a pair of scales, everything on the left of the equal sign is in the left pan of the scales and everything on the right side of the equal sign is in the right pan. The pans are in balance, if anything is added or taken from the left pan then the same needs to be added or taken from the right pan. In our equation above if we take 5 from the left pan we must take 5 from the right pan as well, when 5 is taken from the left pan which has a 5 none is left and when we take 5 from the right pan we have 9 - 5 in the right pan. It thus appears as if the 5 has been moved from the left pan to the right pan with the sign changed. Now go back to the above equation before the 5 was moved and take 9 from both the left and right pans, what does the equation look like now?

Getting back to the simplification process, we have simplified the equation so far to:

x + 2y + 4xy - yx = 4

There is one more pair of terms that can be combined, this is 4xy and -yx. Having the x and y different ways round is not important, xy mean x multiplied by y and multiplication's can be reversed (2 * 3 = 3 * 2). So -yx is the same as -xy, now you can seen the terms are the same and combining the terms gives 3xy. The equation has now become:
x + 2y + 3xy = 4

No more terms can be combined and the equation is now in its simplest form i.e. the original equation has been simplified.

Further explanation to come soon.

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