Combining like terms

If we were sitting at a table with a pile of paper clips, pencils, markers, rubber bands and sticky notes in the center, and if I were to ask you to tell me how many items were in the pile, you would automatically count the things that are alike. For example, you might tell me that there were 5 paper clips, plus 2 markers, plus 9 rubber bands, plus 12 sticky notes. Now, if I were to add two X's made out of stenciled paper, your answer would become: 5 paper clips, plus 2 markers, plus 9 rubber bands, plus 12 sticky notes, plus 2 X's. Now, if I were to add 10 Y's made out of stenciled paper, your answer would become: 5 paper clips, plus 2 markers, plus 9 rubber bands, plus 12 sticky notes, plus 2 X's, plus 10 Y's. The best information we can give about a collection of items is to combine the items that are alike.

When we take away items, we also take them away from items that are the same. So If I were to take away or subtract 5 paper clips, 2 markers, 2 rubber bands, and 7 sticky notes. We would be left with 7 rubber bands, plus 5 sticky notes, plus 2 X's, plus 10 Y's. Next, take away 7 rubber bands and 5 sticky notes. We would have 2 X's and 10 Y's which we can write in shorthand as $$2X+10Y$$.

Suppose I put these paper letters in a pile in the center of the table:


 * X, X, Y, Y, Y, Y, Z, Z, Z, Z, Z, Z.

We have 2 X's plus 4 Y's, plus 6 Z's which we write as $$2X + 4Y + 6Z$$.

Similarly,


 * X, Y, Y, Z, X, Y, Z, Z, Y, Z, Z, Z is also $$2X + 4Y + 6Z$$, because we are combining like items, no matter what the order.

If I were to add 2 Z's to the collection, it would look like X, Y, Y , Z, Z, Z, X, Y, Z, Y, Z, Z which means we need to add $$2Z$$ to the original “shorthand”: $$2X + 4Y + 6Z + 2Z$$, to give us $$2X + 4Y +8Z$$.

Taking two Zs away from the original set leaves


 * X, Y, Y , Z, Z, Z, X, Y, Z, Y. The 6 Z's minus 2 Z's leaves 4 Z's, written as $$2X +4Y +6Z -2Z$$, becomes $$2X + 4Y + 4Z$$.

All we are doing, then, is counting, or combining, items (called terms in mathematics) which are alike. Terms are easy to recognize, they are separated by a plus or minus sign. The directions for combining like terms is to often "Simplify". In fact, "Simplify" means to add, subtract, multiply or divide, but for now, we are simply combining with addition and subtraction.

Some examples
Let's take a look at some examples to see the variety of ways we might combine like terms to simplify an expression.

Simplify the following: (Combine terms that are alike)


 * $$X + X + Y + Y + Y = 2X + 3Y$$


 * $$3X + 2X + 5Y = 5X + 5Y$$


 * $$X + 5Y + 2Y + 3Z + 8Z = X + 7Y + 11Z$$


 * $$5X - 2X +7Y + Y + 9Z - 4Z = 3X + 8Y + 5Z$$

Now, let’s jumble them up like they would be in the center of the table. You look for like terms and combine them just the same.
 * $$3Z + 4X + 2Z + 8Y + 2X + Y = 6X + 9Y + 5Z$$

Now let’s take some off of the table and combine what we have left.
 * $$3Z +4X + 2Z + 8Y +2X +Y -3X -4Y - Z = 3X +5Y + 4Z$$