Combining Like Terms
Patrick is selling items on Craigslist. Among the items he's selling are a novelty shot glass collection and some antique dolls. Antonia sees the listing and messages Patrick: "I'm interested in buying the shot glasses and the dolls. How many do you have?" Patrick responds back "there are 13." At this point, Antonia is confused. Does Patrick mean that he has 13 shot glasses and 13 dolls? Is there a total of 13 items? Are there 13 of one item and he didn't notice that she asked about the other? She emails back and gets clarification: there are 10 shot glasses and 3 dolls. Now Antonia can actually make a decision!
By adding the 10 shot glasses to the 3 dolls, Patrick combined things that were not alike, which caused confusion. On the other hand, if he had 7 shot glasses that he inherited from his grandfather's collection and 3 more that he collected himself, combining those to say "10 shot glasses" would be completely appropriate, since they're all shot glasses.
The same principle applies when combining terms in an equation or expression: we can only combine like terms. If you have $7x+8y$, you can't add them up to say that you have 15 of something, because $7x$ and $8y$ are not the same type of thing. (After all, what would you even call it? What do you have 15 of? It's not $15x$, and it's not $15y$...) However, if you have $4x+18x$, then you can combine them to get $22x$, because they are alike - they are all x's, so you can say you have 22 of them.
So, how do you tell which terms are like terms? The simple answer is this: two terms are like terms if they have the same variables in them. You put x's with other x's, y's with other y's, m's with m's, and constants (terms with no letters) with other constants. Treat the different variables as labels - you can't add x's to y's, any more than you can add dolls to shot glasses or apples to oranges.
For example, consider the expression $17a+4b - 12c +6a - 2 + 29c - 14b + 5 + 18a$. We'd put all the $a$ terms together ($17a +6a +18a$), all the $b$ terms together ($4b-14b$), all the $c$ terms together ($-12c+29c$), and all the constants together ($-2+5$). So it would become $41a-10b+17c+3$, and wouldn't be able to be combined any further because none of the terms we have now are alike.
The place where like terms can get tricky is when you start to have multiple variables in a term, or when you have powers of a variable. The key thing to remember is that terms are only alike if all the variables are exactly the same. If you have $x$, $xy$, and $x^4$, those are not like terms - even though they all have $x$ in them, they don't have the same combination of variables. Remember that $x^4$ means $x$ multiplied by itself four times, or $xxxx$, so it has too many $x$'s to be a like term with $x$! Different powers of the same variable are not like terms.
Also, watch out in situations where fractions are involved. If a particular variable is in the denominator of a fraction on one term, and in the numerator on another term, then those are not like terms. For example, $\frac{4}{x}$ and $\frac{3x}{7}$ are not like terms, because one is an $x$ term and the other is a $\frac{1}{x}$ term. Similarly, $\frac{5jk}{m}$ and $\frac{6jm}{k}$ are not like terms, because the $k$ and the $m$ have switched between the numerators and denominators of the two fractions. On the other hand, $\frac{3x}{2} $, $5x$ and $\frac{1}{4}x$ are all like terms because each one is a constant times x.
"Antique dolls" by stanzebla is licensed under CC BY-SA 2.0
Novelty Shot Glasses image by Ed! at English Wikipedia is licensed under CC BY-SA 3.0
"Apples & Oranges - They Don't Compare"by TheBusyBrain is licensed under CC BY 2.0