Algebraic Expressions - Combining Like Terms

Combining Like Terms

You probably don't keep bananas in your sock drawer or toothpaste on your bookshelf. If you do, we can recommend a good therapist. Aside from it being an awful hassle to run from the bookshelf to the bathroom sink every time you need to spit, the reason we don't do stuff like this is because it makes sense to keep like things together. Algebraic terms are no exception.

A term in an expression has two parts: the constant part, also called the coefficient, and the variable part. Just as they sound, the constant part is the product of all the constants in the term, and the variable part is the product of all the variables in the term. If someone approaches you about Variable Life and Annuities, they probably want your help on an algebra problem.

Sample Problems

1. In the term 9x, the constant part, or coefficient, is 9 and the variable part is x.

2. In the term 2y(3x²), the coefficient is 6 and the variable part is x²y.

3. In the term mc² where c is a constant, c² is the coefficient and m is the variable part.

In an expression, like terms are terms with the same variable part. Going back to one of our earlier examples, bananas go with bananas, and socks go with socks. Although, if bananas could go with socks, we sure could make some hilarious puppets.

Sample Problems

1. The terms x and 8x are like terms, since they both have x as their variable part.

2. Both 5x⁷y and 89x⁷y are like terms, since they both have x⁷y as their variable part.

3. The numbers 4 and 5 are like terms, since both are constants with no variable part. We know what we're getting them for Chanukah.

Non-Sample Problems

1. The terms x and x² are not like terms, since they have different variable parts.

2. The variables y and x are not like terms, since they have different variable parts.

3. The terms x²y and xy² are not like terms, since they have different variable parts.

If we have a pile with 5 lemons and a pile with 6 lemons, we can combine them to get one pile with 11 lemons. Of course, what we actually should do is make lemonade, but that's more of a life philosophy and less of a mathematical solution. In this real-life situation, the lemons are our variables. Therefore, we can combine like terms. Or, in this case, combine like lemons.

Sample Problems

1. Combining 3 homework assignments with 2 homework assignments gives us 5 homework assignments. Plus 1 disgruntled student.

2. Combining 9 fingers and 1 finger gives us 10 fingers. Thank goodness. We thought we misplaced one for a second there.

3. Combining 3 copies of the letter x with 5 copies of the letter x gives us 8 copies of the letter x. With which we can spell the word "xxxxxxxx."

In order to combine terms, they need to be alike. We can't combine x and 2x². We could use the distributive property to rewrite x + 2x² as (1 + 2x)x (give it a try; the math checks out), but (1 + 2x)x isn't that much prettier than x + 2x². Not to sound superficial, but when it comes to mathematical expressions, it's all about the looks.

Sample Problem

Combine like terms in the expression 7x + 3xy + 3 + 4xy + 2x² + 6.

First, rearrange the expression to put like terms next to each other. Pull your bananas out of your sock drawer and move them over to the banana drawer of your dresser. Wait a minute...

7x + (3xy + 4xy) + 2x² + (3 + 6)

Now it's easier to see what goes with what. Which is good, because you don't want your terms to clash.

Combine the terms with the variable part xy first:

7x + (3xy + 4xy) + 2x² + (3 + 6) =
7x + 7xy + 2x² + (3 + 6)

Then combine the constant terms:

7x + 7xy + 2x² + 9

This line is as clean and clear as it gets. Without over-the-counter acne medication, anyway.

Be careful when fractions are involved. Some of them have sharp edges. Using the distributive property to combine like terms with fractional coefficients is a little more complicated, so pay extra close attention to what you're doing. Put on your thinking cap, if you have one. If you don't, a particularly snug baseball cap will do.

Sample Problem

Combine like terms:

First, do some rearranging so that like terms are next to each other:

Next, use the distributive property to factor out the variables.

Finally, add the fractions like we learned to do in the section on fractions. As we're sure you recall, we'll need to convert these guys so that they have like denominators. If you don't remember how to do that, return to Start and draw 2 cards.

Thinking about the distributive property allows us to separate out the coefficients from the variables, which means we don't need to think about both at once. Remember how well it worked when you tried to think at the same time about riding your bicycle and about that kiss you got from Heather Pelnicki? Didn't turn out well for you, did it? Didn't turn out well for Heather Pelnicki, either. To be fair, she shouldn't have been standing right in the middle of the bike lane.

• Eliminating Parentheses

  Getting Rid of Parentheses—Who Needs 'Em?

  We've seen a lot of expressions with parentheses in them on this algebraic journey of ours. We prefer them as smiley faces in emoticons—the rest of the time, they sort of confuse things. Here's a valid, but horrid, expression:

  ((x + 1)((3 – 4x)(4x + 2) – 5))²

  While this expression makes mathematical sense, the high number of parentheses in it makes it hard to tell what's going on. In terms of English, how would you like to read this sentence?

  "This sentence (the one you are reading (as if we would be referring to any other sentence) right now (as opposed to later)) is so (poorly) constructed (put together (like a building with an unstable foundation)), it makes me (literally) want to vomit."

  Not pretty, right? To make things clearer, we can use the symbols { } or [ ] in place of some of the parentheses ( ). While still not the most pleasant-looking expression we've ever seen, at least it's a little easier to figure out what's being multiplied by what in this rewritten expression:

  [(x + 1){(3 – 4x)(4x + 2) – 5}]²

  It is okay to think of ( ), { }, and [ ] all as parentheses, since they're all used for the same purpose. Because these symbols serve to group terms together, we also call them grouping symbols. The person who uses these symbols is called the grouper. He tastes delicious when broiled and seasoned with ground white pepper and paprika.

  When we're given an expression with a plethora of parentheses, one way to rewrite the expression is to eliminate parentheses until they're all gone. Unfortunately, there's no special "Parentheses-B-Gone" spray you can buy at CVS. Getting rid of parentheses often involves using the distributive property. 

  Sample Problem

  Eliminate parentheses in the expression 4{x + 2(3 – x²)}.

  First way: use the distributive property on the inner set of parentheses to get 4{x + 6 – 2x²}. Then distribute the remaining set of parentheses to get 4x + 24 – 8x². Parentheses, we hardly knew ye.

  Second way: use the distributive property on the outer set of parentheses to arrive at 4x + 8(3 – x²), then on the remaining set of parentheses to get 4x + 24 – 8x². Reassuringly, we got the same answer as we did when we tried it the first way.

  Third way: yell "Fire!" and watch them scatter.

  As long as we're careful with the arithmetic, it's okay to get rid of parentheses in any order. In fact, it provides a good way to check our work. No matter the order in which we eliminate parentheses, we should get the same answer. We can solve each problem a couple of different ways to make sure we're on the right track. We figured you'd probably want to solve each problem more than once anyway, so this situation works out perfectly.

  Be Careful with negative signs, and remember that, to distribute a negative sign over a quantity in parentheses, we erase the negative sign and parentheses while flipping the signs of all the terms in the parentheses. You'll want to flip them about once every thirty seconds to ensure maximum crispiness.

• Simplifying

  Simplifying: Putting it All Together

  To simplify an expression means to tidy it so that it has as few parentheses and as few terms as possible. Ideally, it would be nice if the entire thing vanished into oblivion. That would make our lives much easier.

  The way we simplify is by eliminating parentheses and combining like terms until there are no parentheses left and no like terms left to combine. Break out that vacuum cleaner.

  Sample Problem

  Simplify 4{3 + (x – 1)}.

  First way: work from the inside out. We don't actually need those innermost parentheses. Get rid of them. They're like wisdom teeth or a Blockbuster Video gift card: completely useless and unnecessary.

  4{3 + x – 1}

  Now combine like terms to get:

  4{2 + x}

  Finally, multiply using the distributive property:

  8 + 4x

  Second way: work from the outside in. First, use the distributive property to get:

  12 + 4(x – 1)

  Then, use the distributive property again:

  12 + 4x – 4

  And finally, combine like terms:

  8 + 4x

  Okay, we're all combined out now. Break time.

  Be Careful: Simplifying stuff in parentheses before you use the distributive property will often save you work. It's also a good exercise in applying the Order of Operations we learned before. Remember Please Excuse My Dear Aunt Sally? Notice that, in the second way of solving the example above, the distributive property needed to be used twice instead of once. But, you know what they say: "Using the distributive property twice is just as nice!" We're not exactly sure whom the "they" is referring to, but somebody somewhere must say it. 

  Sample Problem

  Show that the area of the trapezoid pictured below is given by 

  

  First, let's chop the trapezoid into pieces (aw, don't feel bad—it has no nerve endings), and re-label some lengths. The lower base of the trapezoid has length b₂, which we can rewrite as x + b₁ + (b₂ – b₁ – x).

  

  The area of the trapezoid is the area of the triangle on the left side, plus the area of the rectangle, plus the area of the triangle on the right side. Using the area formulas for triangles and rectangles, we get:
  

  

  Ay, chihuahua! We sure would love to simplify that bad boy. Applying the distributive property gets us:

  

  Now combine like terms, remembering that b₁h = hb₁:

  

  Hey, look at that. Those hx terms canceled each other right out. For the final step, factor out to get:

  

  Couldn't be more straightforward. Well, it could, but then it wouldn't be as much fun.