Combining Like terms

When children first learn about mathematics, they usually begin by counting objects. For example they may begin to learn by counting apples. Once they become proficient in the process they move away from counting objects to the abstract idea of numbers. Combining like terms puts both of these ideas together. If you have 5 apples and   someone gives you 2 apples, then you have 7 apples in total. If you have 5 apples and someone gives you 4 oranges, then you have 5 apples in total. The same concepts applies to combining like terms. Terms are expressions that can contain numbers and letters that are separated by plus (+) or minus (-) signs. Here are some terms; 5x, 2xyz, 54rs, 65cd, x. Just like the apples and oranges the number in front of the letters indicate the amount of those particular items. There are 5 x's, 2 xyz's, 54 rs's, 65 cd's, and a single x. The single x and the 5x are like terms that can be combined to obtain a total of 6 x's.
5x + x = 6x

So combining like terms is really about condensing expressions by adding or subtracting similar terms. Sometimes people make the mistake of writing 6x^2 (x with an exponent of 2), which implies that there are 6 x^2's instead of 6 x's. The number in front of the variable is just shorthand notation for the amount of these variables in our case we write 6x instead of writing x,x,x,x,x,x. To help keep from changing the variable when combining like terms, think of combining like terms as counting people at a gathering by their last names. Here is a list of some people from the Gray family reunion.
Bobby Gray
Tommy Gray
Sandy John
Rodney John
Susan Young
Anthony Gray
Stacey Gray
Tonya Sullivan
Grant Sullivan

The last name break down is 2 from the Gray family, 2 from the John family, 1 from the Young family, 2 more from the Gray family, and 2 from the Sullivan family.
So this is
2 Gray + 2 John + 1 Young + 2 Gray + 2 Sullivan  = 4 Gray + 2 John + Young + 2 Sullivan

If we use the first letter of each last name to represent the groups we would have
2G + 2J + 1Y + 2G + 2S = 4G + 2J + Y + 2S

More examples.
23xy + 12xy = 35xy

3x - 5y - 4z + 4x + y + 6z = x - 4y + 2z

4x^2 + 2x - 7x^2 = -2x^2 + 2x

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