Have you ever had your picture taken at an amusement park? They have those old time costumes that you can put on and everyone stands together for a photo?

Well, at the amusement park, Kelly has decided to gather a group of friends to do just that. When she first posed the question at lunch, she had five people say that they wanted to do it.

Then later in the day, four more people wanted to join it.

There is a fee per person if you want to be in the picture.

Kelly wants to figure out the total number in simple way and include the fee. How can she do this?

To do this, Kelly can write an expression using a single variable.

Do you know how to do this?

**This Concept will show you how to write single variable expressions that include sums and differences. Then you will understand how to help Kelly.**

### Guidance

In the last few Concepts, you learned how to write single-variable expressions and single variable equations. Now you are going to learn to work with single-variable expressions. The first thing that you are going to learn is how to ** simplify** an expression.

**What does it mean to simplify?**

To simplify means to make smaller or to make simpler. When we simplify in mathematics, we aren’t solving anything, we are just making it smaller.

**How do we simplify expressions?**

Sometimes, you will be given an expression using variables where there is more than one term. A term is a number with a variable. Here is an example of a term.

\begin{align*}4x\end{align*}

This is a term. It is a number and a variable. We haven’t been given a value for \begin{align*}x\end{align*}, so there isn’t anything else we can do with this term. It stays the same. If we have been given a value for \begin{align*}x\end{align*}, then we could evaluate the expression. You have already worked on evaluating expressions.

When there is more than one LIKE TERM in an expression, we can simplify the expression.

What is a like term?

A like term means that the terms in question use the same variable.

**\begin{align*}4x\end{align*} and \begin{align*}5x\end{align*} are like terms. They both have \begin{align*}x\end{align*} as the variable. They are alike.**

**\begin{align*}6x\end{align*} and \begin{align*}2y\end{align*} are not like terms. One has an \begin{align*}x\end{align*} and one has a \begin{align*}y\end{align*}. They are not alike.**

We can simplify expressions with like terms. We can simplify the sums and differences of expressions with like terms. Let’s start with sums.

\begin{align*}5x+7x\end{align*}

First, we look to see if these terms are alike. Both of them have an \begin{align*}x\end{align*}, so they are alike.

Next, we can simplify them by adding the numerical part of the terms together. The \begin{align*}x\end{align*} stays the same.

\begin{align*}&5x+7x\\ &12x\end{align*}

You can think of the \begin{align*}x\end{align*} as a label that lets you know that the terms are alike.

\begin{align*}7x+2x+5y\end{align*}

First, we look to see if the terms are alike. Two of the terms have \begin{align*}x\end{align*}’s and one has a \begin{align*}y\end{align*}. The two with the \begin{align*}x\end{align*}’s are alike. The one with the \begin{align*}y\end{align*} is not alike. We can simplify the ones with the \begin{align*}x\end{align*}’s.

Next, we simplify the like terms.

\begin{align*}7x+2x=9x \end{align*}

We can’t simplify the \begin{align*}5y\end{align*} so it stays the same.

\begin{align*}9x+5y\end{align*}

**This is our answer.**

We can also simplify expressions with differences and like terms.

\begin{align*}9y-2y\end{align*}

First, you can see that these terms are alike because they both have \begin{align*}y\end{align*}’s. We simplify the expression by subtracting the numerical part of the terms.

9 - 2 = 7

**Our answer is \begin{align*}7y\end{align*}.**

Sometimes you can combine like terms that have both sums and differences in the same problem.

\begin{align*}8x-3x+2y+4y\end{align*}

We begin with the like terms.

\begin{align*}8x-3x&=5x \\ 2y+4y&=6y\end{align*}

Next, we put it all together.

\begin{align*}5x+6y\end{align*}

**This is our answer.**

Remember that you can only combine terms that are alike!!!

*Use your notebook and pencil to take some notes on how to identify like terms.*

Try a few of these on your own. Simplify the expressions by combining like terms.

#### Example A

\begin{align*}7z+2z+4z\end{align*}

**Solution: \begin{align*}13z\end{align*}**

#### Example B

\begin{align*}25y-13y\end{align*}

**Solution: \begin{align*}12y\end{align*}**

#### Example C

\begin{align*}7x+2x+4a\end{align*}

**Solution: \begin{align*}9x + 4a\end{align*}**

Here is the original problem once again.

Well, at the amusement park, Kelly has decided to gather a group of friends to do just that. When she first posed the question at lunch, she had five people say that they wanted to do it.

Then later in the day, four more people wanted to join it.

There is a fee per person if you want to be in the picture.

Kelly wants to figure out the total number in simple way and include the fee. How can she do this?

To do this, Kelly can write an expression using a single variable.

Do you know how to to do this?

Now we can use the information that is provided in the dilemma to write a single variable expression.

First, 5 people wanted to be in the picture.

There is also a fee per person. Kelly doesn't know this amount. It is our variable \begin{align*}x\end{align*}

\begin{align*}5x\end{align*}

Then 4 more people wanted to join the picture. The fee per person applies to them too.

\begin{align*}5x + 4x\end{align*}

If we combine like terms, \begin{align*}9x\end{align*} is the expression Kelly can use to figure out the total cost of the picture. Once she knows the cost per person, she will be able to substitute that into the given expression and solve for the total cost.

But wait, this is information we can use in another Concept.

### Vocabulary

Here are the vocabulary words in this Concept.

- Expression
- a combination of variables, numbers and operations without an equal sign.

- Simplify
- to make smaller

### Guided Practice

Here is one for you to try on your own.

\begin{align*}5x + 2x - 1x + 6y - 4y\end{align*}

**Answer**

To simplify this expression, we simply combine the terms that are alike.

\begin{align*}5x + 2x - 1x = 6x\end{align*}

\begin{align*}6y - 4y = 2y\end{align*}

Now we put those simplified terms together.

\begin{align*}6x + 2y\end{align*}

**This is our answer.**

### Video Review

Here is a video for review.

James Sousa: Simplifying Algebraic Expressions

### Practice

Directions: Simplify the following expressions by combining like terms. If the expression is already in simplest form please write “already in simplest form.”

1. \begin{align*}4x+6x\end{align*}

2. \begin{align*}8y+5y\end{align*}

3. \begin{align*}9z+2z\end{align*}

4. \begin{align*}8x+2y\end{align*}

5. \begin{align*}7y+3y+2x\end{align*}

6. \begin{align*}9x-x\end{align*}

7. \begin{align*}12y-3y\end{align*}

8. \begin{align*}22x-2y\end{align*}

9. \begin{align*}78x-10x\end{align*}

10. \begin{align*}22y-4y\end{align*}

11. \begin{align*}16x - 5x + 1x - 12y + 2y\end{align*}

12. \begin{align*}26x - 15x + 12x - 14y + 2y\end{align*}

13. \begin{align*}36x - 5x + 11x - 1x + 2y\end{align*}

14. \begin{align*}26x - 25x + 12x - 13y + 2y\end{align*}

15. \begin{align*}29x - 25x + 18x - 12x + 12y + 3y\end{align*}