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Combining Like Terms in Algebraic Expressions

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  • 0:06 Puzzles
  • 0:33 Combining Like Terms
  • 1:51 Adding Variables
  • 3:32 Practice Problems
  • 6:40 Lesson Summary
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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.

Puzzles

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.

Combining Like Terms

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 98y.

Let's connect 'like' and 'terms.' Like terms are individual terms that have the same variable. For example, 3x, 95x and 17x are all like terms. They all have a single variable, x. 4y^2 and 12y^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.

Adding Variables

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 + 5x, we can add those to get 6x. But if we had x^2 + 5x, 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 + 5x would be 2 + 5*2, which is 2 + 10, or 12. When we added x + 5x, we got 6x. If x = 2, what is 6x? It's still 12.

But what about x^2 + 5x? If x = 2, x^2 + 5x is 2^2 + 5*2. That's 4 + 10, or 14. How could you combine x^2 and 5x? Would it be 6x^2? Well, then if x = 2, 6x^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 xs. But we can't add different variables, like x + y. x + y does not equal 2x and it doesn't equal 2y. When we have different variables, that means they're potentially representing different numbers.

Practice Problems

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

Here's one: x^2 + 2 + 6x^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 + 6x^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 7x^2. That makes our simplified expression 7x^2 + 5.

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