In order to combine like terms, you will often need to use multiple properties. In this video, you will use a combination of the commutative, associative and distributive properties together to solve the example presented.
In the problem shown--22 times the quantity x-5 plus 33 minus 6x--we’re going to use several properties. We’re going to use the distributive property, the commutative property, and the associative property. So to begin with a problem like this, the directions will basically say to simplify. Eventually you’ll be solving for equations using these phrases or expressions, but for now simplification is all you can do, which means you’re going to combine your terms and get it to not be such a long string of terms. So we’re going to start by distributing. The first thing we do, if we think back to our order of operations, PEMDAS, which is "People Eat Monkeys Dogs and Snakes" or "Please Excuse My Dear Aunt Sally," you’re going to distribute the 22 in front of the parenthesis first. So we’re going to take 22 times x and 22 times 5 and distribute that. So I’m going to start with 22 times x. I have a minus sign, so I’m going to bring that down and 22 times 5 is (what is that?) 110. And I’m going to just bring down the rest of the terms. So I like to write it out in steps, so that I can find mistakes if there are any as I go, and this way it helps us to visualize and see we’re not losing negatives in our minds as we do this.
The next step we’re going to use is to rearrange our problem. We can use the commutative property and rearrange or swap out numbers. I’m going to change this so that I don’t miss the negatives. I’m going to change this minus to a plus, a negative, and this subtraction to a plus, a negative, so that I can easily use my commutative property. If you take a look at what we have here, I have 22x and I have -6x. They go together. They’re like terms. You might want to think of this as, you know, those little money coin holders that you get from the bank? You’ve got quarters, you’ve got dimes, you’ve got nickels. . . . they’re all different sizes and they have different coin holders so you can group them together like coins. Think of that. . . . think of the terms as a coin. 22x is a different size than -110 or as a plain number. So the numbers you are going to put together and your variables that have the same exponent you’re going to put together. So I’m going to use my commutative property to basically swap out my numbers, so if I do 22x plus a -6x, and then I do -110 plus 33, you’re actually combining that with the associative property as well. So I can take my commutative property and swap numbers in order, but I also, if you notice, am regrouping them, so I’m using my associative property by regrouping . . . oops . . . the numbers together. So when I do that I’m going to combine my terms here -6x plus 22x, and I end up with (let’s see) 22 minus 2 is 20 minus another 4 is 16x and that’s what I get in the first set of parenthesis. And I have a plus sign here. A -110 plus 33 is -77. So my simplification is 16x minus 77.
You Try It!
Simplify: -2(3x + 5) - 6x + 27
-6x + (-10) + (-6x) + 27
[-6x + (-6x)] + [(-10) + 27]
-12x + 17
