Pre-Algebra: Simplifying Algebraic Expressions

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COMBINING LIKE TERMS AND USING THE DISTRIBUTIVE PROPERTY

Let’s see if we can combine like terms, take complicated algebraic expressions, and make them simple. Let’s start off easily. 9a plus 11a. These are like terms, so we can combine them. One way to think of them in your mind, which might be helpful, is I’ve got 9 a’s and I’m adding on 11 more a’s. How many as do I have in total? You have 9 + 11 a’s, or 20 a’s. This can be simplified to 20a. Since they’re like terms, I just take the coefficient and perform the addition. 9 plus 11 is 20, and the a’s remain. That’s all there is to that.

Let’s make it a lot harder, a lot quicker. I not only want us to simplify this but also talk about some of the rules required to simplify it. We’ll see a lot of arithmetic properties we’ve talked about before. I’ve got 4w plus 7x squared minus 3w plus z plus 3x squared plus 5z minus 6x. It’s so much stuff. I want to combine like terms. How do I do that? Notice that here I’ve got ws. They can be combined, but there are steps involved in combining them. Let me talk through them with you. See they’re not next to each other, so I can’t put them together.

The first thing I’d have to do is use the Associative Property to associate or put parentheses around these two terms, and say I want to perform that addition first. Looking inside the first two terms, I can switch these using the Commutative Property of Addition. I switch them. When I switch them you see 7x squared plus 4w and then I see minus 3w, and then I’ve got these other terms. Notice these terms are close to each other, so I can combine them once I use the Associative Property again. What do I do here? I can take these two and use the Associative Property to clump those two together. Then I can use the Commutative Property to switch the order, so I see 3x squared plus z, and now the zs are next to each other. I’m going to do that now. I’ll write that as plus 3x squared plus z plus 5z minus 6x. See how I clumped these together and switched their order? Now, I can use the Associative Property to write this as this. When you write all this out it looks more complicated. It’s about to get really simple really fast. These are like terms. 4w minus 3w is w. 7x squared plus w plus 3x squared, and I can perform this addition. z plus 5z—how many total zs? I have 6 of them. 6z minus 6x. We might be tempted to say these are like terms because the 6’s are the same. Those are the coefficients. Remember to look at the variables—x and z are not equal. These stay the same. We cannot combine these. Is there any other simplification I can do? Yeah. I could use the Associative Property to put these two things together. Then once they’re associated I can use the Commutative Property to switch the order. Then I’m going to have a lot of like terms next to each other. 7x squared plus 3x squared plus w plus 6z minus 6x. I clumped those two things together, and using the Commutative Property I switched them. Now I can use the Associative Property to clump these two. Notice I have 7x squared and 3x squared here. So 10x squared plus w plus 6z minus 6x. Can I combine anything else? The 6’s can’t be combined, because these variables are different, w is by itself, no other x squared, no more zs or xs, so that’s simplified as much as possible. That’s our simplified version of this algebraic expression. This is the same algebraic expression, but this is a little bit more simple. Wow.

Let’s try one last one. I want us to simplify this expression. 5 times the quantity z plus 3z squared plus 5x. It’s 5 times a quantity, so the first thing I should do is use the Distributive Property to take the 5 and distribute it, multiply it by the z and the 3z squared. I get rid of the parentheses by using the Distributive Property. Let’s do that now. When I do that, I see 5z plus 15z squared plus 5x. Great. Now you can imagine a friend saying: we can simplify more because the 5’s are the same. What would you say to them? You’d say no—those are the coefficients, not the variables. This is a z and this is an x. They can’t be combined. Another friend might say we can put the z and the z squared together. No—z and z squared are different. You can only combine zs with zs and z squared with z squared—only combine like terms. You’ve got to be careful, but you can simplify algebraic expressions. That makes our work in the future a little simpler.