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ELM: CSU Math Study Guide16 chapters | 140 lessons

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Lesson Transcript

Instructor:
*Jeff Calareso*

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

When you have an algebraic expression that's much too long, it would be great if you could simplify it. That's when knowing how to combine like terms comes in. In this lesson, we'll learn the process of combining like terms and practice simplifying expressions.

I love jigsaw puzzles. There's something very gratifying about first finding all the edge pieces, then building that border. Then I like to sort the other pieces by color. Maybe it's a landscape painting, and a good third of the top of the puzzle is all blue sky. If I can't differentiate the pieces by color, I'm matching by shape. One way or another, the picture reveals itself. Building a jigsaw puzzle is very much like combining like terms with algebraic expressions.

The phrase 'combining like terms' is kind of a puzzle itself. Let's look at the pieces of that phrase and then put them together.

First, let's talk about terms. You're probably familiar with things like constants, which are just ordinary numbers. And then there are variables, like *x* or *y*. These are just symbols used in place of numbers we don't yet know. When you start putting these together, you get terms.

In algebra, **terms** are constants, variables and products of constants and variables. So terms can be 1, 38 and 356.3. They can also be *x*, *a*^2 or 98*y*.

Let's connect 'like' and 'terms.' **Like terms** are individual terms that have the same variable. For example, 3*x*, 95*x* and 17*x* are all like terms. They all have a single variable, *x*. 4*y*^2 and 12*y*^2 are also like terms since they share the *y*^2. Constants are also considered like terms. These are numbers without variables, like 2, 5.4 and -188.

Our puzzle is almost complete. Let's add 'combining' to 'like terms.' **Combining like terms** is the process of simplifying expressions by joining terms that have the same variable.

You know you can add 2 + 3. That's a form of combining like terms. When we have variables, we do much the same thing. The key is to pay attention to the exponent. You can only add variables if they have the same exponent.

For example, if we have *x* + 5*x*, we can add those to get 6*x*. But if we had *x*^2 + 5*x*, we couldn't add those.

Why not? Remember, the variable is just a symbol. In our example, *x* is standing in for a number we don't know. What if *x* = 2? Then *x* + 5*x* would be 2 + 5*2, which is 2 + 10, or 12. When we added *x* + 5*x*, we got 6*x*. If *x* = 2, what is 6*x*? It's still 12.

But what about *x*^2 + 5*x*? If *x* = 2, *x*^2 + 5*x* is 2^2 + 5*2. That's 4 + 10, or 14. How could you combine *x*^2 and 5*x*? Would it be 6*x*^2? Well, then if *x* = 2, 6*x*^2 would be 6(2)^2, which is 6*4, or 24. 24 doesn't equal 14.

Also, note that we can only add similar variables, like if we have two *x*s. But we can't add different variables, like *x* + *y*. *x* + *y* does not equal 2*x* and it doesn't equal 2*y*. When we have different variables, that means they're potentially representing different numbers.

Okay, let's try some practice problems and get comfortable with combining like terms.

Here's one: *x*^2 + 2 + 6*x*^2 + 3. Think of this like a jigsaw puzzle. We need to put the like terms together. Which terms are like each other? First, we have two numbers with no variables: 2 and 3. Those can be combined to give us 5. So now we have *x*^2 + 6*x*^2 + 5. And those *x* terms - do they have the same exponent? They do. They're both *x*^2. So we can add them to get 7*x*^2. That makes our simplified expression 7*x*^2 + 5.

Let's try another: 2*x*^2 - 5*x* + 3*x* + 8*x*^2 + *y*. Okay, again, put the puzzle pieces together. That 5*x* and 3*x* share the same exponent, so let's combine those. But wait - don't forget that that 5*x* is really -5*x*, so -5*x* + 3*x* = -2*x*. And that 2*x*^2 and 8*x*^2 both have an *x*^2 in them, so we can combine them to get 10*x*^2. What about that *y*? There are no other terms with a *y* in them, so we can't do anything with that. That means our simplified expression is 10*x*^2 - 2*x* + *y*. Okay, it's not a landscape painting or a picture of cats playing with yarn, but it is a simpler expression than what we started with.

Let's try a different kind of algebraic expression: [(-5*y* + 8*y*) - (6*y* + 2)] - [(3*y* -*y*) + 9*y*]. The trick with this one is to not lose track of those negative signs. Let's start with what's inside parentheses. We can combine this first -5*y* and 8*y* to get 3*y*.

Now, what about that 6*y* and 2? They're not like terms, so we can't combine them. If we distribute the minus sign across the parenthesis, this first section becomes 3*y* - 6*y* - 2.

Okay, there's more we can do there. We can combine the 3*y* and 6*y* - remember that it's a -6*y* - and get -3*y* - 2.

Next, let's look at the second part. We can combine 3*y* and -*y* to get 2*y*. And we can add that 2*y* to this 9*y* to get 11*y*.

It's time to bring these two sections together. So we have -3*y* - 2 and 11*y*. But remember that there's this minus sign before the 11*y*. So it's -3*y* - 2 - 11*y*.

Anything else we can combine? Yep, the -3*y* and -11*y*. That becomes -14*y*. So -14*y* - 2 is our final expression.

We took an expression with 7 terms and, by combining like terms, got it down to just 2. That's pretty good for matching puzzle pieces!

In summary, combining like terms is just the process of simplifying expressions by joining terms with the same variable. Terms include constants, or numbers, and variables, like *x* or *y*. We can only add variables if they have the same exponent, and we can't add different variables together.

After finishing this lesson, you should understand the process of combining like terms when solving algebraic expressions.

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ELM: CSU Math Study Guide16 chapters | 140 lessons

- What is a Variable in Algebra? 5:26
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