6 Combining Like Terms in Algebraic Expressions

6 Combining Like Terms in Algebraic Expressions

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.

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.

Let's try another: 2x^2 - 5x + 3x + 8x^2 + y. Okay, again, put the puzzle pieces together. That 5x and 3x share the same exponent, so let's combine those. But wait - don't forget that that 5x is really -5x, so -5x + 3x = -2x. And that 2x^2 and 8x^2 both have an x^2 in them, so we can combine them to get 10x^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 10x^2 - 2x + 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: [(-5y + 8y) - (6y + 2)] - [(3y -y) + 9y]. 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 -5y and 8y to get 3y.

Now, what about that 6y 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 3y - 6y - 2.

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

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

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

Anything else we can combine? Yep, the -3y and -11y. That becomes -14y. So -14y - 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!

Lesson Summary

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.