# How to Add & Subtract Monomials

## Monomials

In this lesson, we will talk about working with monomials. **Monomials** are single terms consisting of a coefficient multiplied with variables. Your **coefficient** is the number that is multiplied by a variable, and your **variable** is a letter representing an unknown value.

For example, in the monomial 4*x*, the 4 is the coefficient and the *x* is the variable. This particular monomial only has one variable, but it is also possible to have more than one variable, such as in 3*xy*, where 3 is the coefficient and *x* and *y* are the variables.

It is also possible for your variables to have exponents. 8*x*^2, for example, is a monomial where 8 is the coefficient and the *x*^2 is the variable *x* with an exponent of 2. The exponent tells you how many you have of the variable. So *x*^2 tells you that you have 2 *x*'s. If your monomial only has variables, such as in *x*, then your coefficient is automatically a 1.

## Only Like Terms

Now that you know what monomials are, the next most important thing to know is that you can only add or subtract like terms. **Like terms** are monomials that share the same variables. For example, 4*x* and 7*x* are like terms because the variable *x* is the same in both. The variables have to be exactly the same. 4*x* and 4*x*^2 are not like terms because their variables are not exactly the same. The coefficients do not have to be the same.

## Adding Monomials

Now, let's see about adding monomials together. Let's say you are given the problem 4*x* + 9*x*. The first thing you need to do is to check to see if your monomials are like terms. You ask yourself, are the variables exactly the same?

Yes, they are. So that means you can go ahead and add them together. To add them together, you leave the variables as they are and you add up the coefficients. So you can add 4 + 9. What does this equal? It equals 4 + 9 = 13. So our answer is 13*x*. We have the variables exactly the same.

If you are given a problem where you don't have like terms, you will leave the terms that are not alike alone. For example, if your problem involved adding several monomials together as in 2*xy* + 5*xy* + 6*y*, you again first look to see if you have like terms. You see that the 2*xy* and the 5*xy* are like terms so that means you can add those two together.

You remember that you leave the variables alone and add the coefficients. You add 2 + 5 = 7; that means 2*xy* + 5*xy* = 7*xy*. But you still have the 6*y*. What do you do with this one? Because this is not a like term, you can't do anything with it. So, you leave it as it is and your complete answer is 2*xy* + 5*xy* + 6*y* = 7*xy* + 6*y*. You have added the like terms and everything else is left alone and included in the answer.

## Subtracting Monomials

The same rules apply when subtracting monomials. You can only subtract those monomials that have like terms. Just like with addition, you leave the variables alone and subtract the coefficients. Also, if you are subtracting monomials with different variables, those that are not like terms, then you subtract the like terms and you leave the other terms alone, but include it in your answer.

So 10*y* - 7*y* becomes 3*y*. We have subtracted the coefficients 10 - 7 = 3, and we have left the variable the same. We were able to subtract these monomials because they are like terms.

However, if we had 8*x* - 8*y*, our answer would be 8*x* - 8*y* because there are no like terms that we can subtract. We need to include all the terms that are not alike in our answer.

If you are given a problem that includes both addition and subtraction, you do the same. You look for like terms that you can add or subtract. You leave the variables alone and you add or subtract the coefficients.

## Lesson Summary

Let's review what we've learned now. **Monomials** are single terms consisting of a coefficient multiplied with variables. Your **coefficient** is the number that is multiplied by a variable, and your **variable** is a letter representing an unknown value.

You can only add or subtract like terms. **Like terms** are monomials that share the same variables. To add or subtract monomials that are like terms, you leave the variables as they are and you add or subtract the coefficients. If you have monomials that are not like terms, you add or subtract as many like terms as you can and you leave the terms that are not like terms in your answer.

## Memorization Table

Monomials |
single terms consisting of a coefficient multiplied with variables |

Coefficient |
the number that is multiplied by a variable |

Variable |
a letter that represents an unknown value |

Like terms |
monomials that share the same variables |

## Learning Outcomes

By studying and completing the practice exercises in this lesson, you can prepare to:

- Define monomials
- Describe what coefficients and variables are
- Explain how to to add or subtract monomials

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