# 4.6: Simplify Variable Expressions Involving Integer Addition

**At Grade**Created by: CK-12

Have you ever been diving? Have you been to the Caribbean?

Last year, on his first day in the Caribbean, Cameron did his dive test and passed with flying colors. A dive test is done in a pool prior to diving. It lets the dive master know that you understand what you are doing and can handle yourself under the water. Scuba diving is exciting, but you have to know what you are doing to do it well.

The second day, Cameron and his Dad went for their first two dives. On the first dive, Cameron traveled to a depth of 25 feet. Then he and his Dad saw a stingray and followed it for a while and traveled down another 10 feet. Cameron took a few pictures of the stingray too.

Then they traveled back to the boat for some surface time to eat and rest before going on the second dive. On his second dive, Cameron did a shallow dive of only 15 feet. He loved seeing the beautiful coral and even spotted a sea cucumber.

When they returned to the boat, Cameron began calculating his total depth and his total time for the day.

**To calculate Cameron’s total depth for the day, you will need to know how to add integers. This Concept will give you all that you need to know!!**

### Guidance

An ** expression** is a number sentence that contains numbers and operations. An expression is not solved it is evaluated because there isn’t an equals sign in an expression. Some expressions include

**. These expressions are called**

*variables***. A**

*variable expressions***is a symbol or letter that is used to represent one or more numbers.**

*variable*
**Variable expressions can be combined when there is a common term or a like term.**

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

**We can simplify or combine this variable expression because it has two common variables. The \begin{align*}x\end{align*}’s are common so we can find the sum of the variable expression.**

**The answer is \begin{align*}12x\end{align*}.**

\begin{align*}9y+8x\end{align*}

**This variable expression can not be combined or simplified. It does not have like terms. The \begin{align*}x\end{align*} and the \begin{align*}y\end{align*} are different, so we can not do anything with this expression. It is in simplest form.**

**Sometimes, expressions will include both variables and integers. You can use what you know about how to add integers to help you find the value of expressions with variables.**

Find the sum \begin{align*}-4a+(-a)\end{align*}.

**Since \begin{align*}-4a\end{align*} and \begin{align*}-a\end{align*} both have the same variable, they are like terms. Use what you know about how to add integers to help you add the terms.**

Both terms have the same sign, a negative sign. So, find the absolute values of both integers. Then add those absolute values to combine the terms. Remember, \begin{align*}-a=-1a\end{align*}.

\begin{align*}|-4|=4\end{align*} and \begin{align*}|-1|=1\end{align*}, so add \begin{align*}4a+1a=5a\end{align*}.

Since both terms had negative signs, give the answer a negative sign.

**The sum of \begin{align*}-4a+(-a)\end{align*} is \begin{align*}-5a\end{align*}.**

Find the sum \begin{align*}-3t+9t\end{align*}.

**Since \begin{align*}-3t\end{align*} and \begin{align*}9t\end{align*} both have the same variable, they are like terms. Use what you know about how to add integers to help you add the terms.**

Both like terms have different signs. So, find the absolute values of both integers. Then subtract the term whose integer has the lesser absolute value from the other term.

\begin{align*}|-3|=3\end{align*} and \begin{align*}|9|=9\end{align*}, so subtract: \begin{align*}9t-3t=6t\end{align*}.

Since \begin{align*}9>3\end{align*}, and \begin{align*}9t\end{align*} has a positive sign, give the answer a positive sign.

**The answer is \begin{align*}-3t+9t=6t\end{align*}.**

**Working with variable expressions may seem tricky, but if you first determine if you have like terms and then use the strategies you have learned for finding integer sums, you will be able to simplify each expression.**

Simplify each variable expression.

#### Example A

\begin{align*}-8x+ -5x\end{align*}

**Solution:\begin{align*}-13x\end{align*}**

#### Example B

\begin{align*}-19y+5y\end{align*}

**Solution:\begin{align*}-14y\end{align*}**

#### Example C

\begin{align*}-6y+2y+ -3y\end{align*}

**Solution:\begin{align*}-7y\end{align*}**

Here is the original problem once again.

Last year,on his first day in the Caribbean, Cameron did his dive test and passed with flying colors. A dive test is done in a pool prior to diving. It lets the dive master know that you understand what you are doing and can handle yourself under the water. Scuba diving is exciting, but you have to know what you are doing to do it well.

The second day, Cameron and his Dad went for their first two dives. On the first dive, Cameron traveled to a depth of 25 feet. Then he and his Dad saw a stingray and followed it for a while and traveled down another 10 feet. Cameron took a few pictures of the stingray too.

Then they traveled back to the boat for some surface time to eat and rest before going on the second dive. On his second dive, Cameron did a shallow dive of only 15 feet. He loved seeing the beautiful coral and even spotted a sea cucumber.

When they returned to the boat, Cameron began calculating his total depth and his total time for the day.

**To help Cameron calculate his total depth, we can write the following equation.**

**Depth of dive 1 + depth of dive 2 = total depth**

**Remember that depth is below the surface, so we use negative numbers to represent these integers.**

**On dive 1, Cameron went -25 ft and then -10 ft.**

**On dive 2, Cameron went -15 ft.**

**Now we can substitute these values into the equation.**

\begin{align*}-25 + -10 + -15 = -50\end{align*}

**Cameron’s total depth for the day was -50 feet.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Integer
- the set of whole numbers and their opposites.

- Sum
- the answer in an addition problem.

- Expression
- a phrase using numbers and operations.

- Variable
- a letter used to represent an unknown quantity.

- Variable Expression
- A phrase using numbers, operations and variables.

### Guided Practice

Here is one for you to try on your own.

Simplify \begin{align*}7z+(-3z)\end{align*}.

**Answer**

We combine like terms in this problem, but keep in mind that there are positive and negative values.

\begin{align*}7z + (-3z)=4z\end{align*}

**This is our answer.**

### Video Review

Here is a video for review.

- This is a James Sousa video on evaluating variable expressions.

### Practice

Directions: Simplify each variable expression.

1. \begin{align*}7z+(-3z)\end{align*}

2. \begin{align*}17z+(-15z)\end{align*}

3. \begin{align*}5x+(-3x)\end{align*}

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

5. \begin{align*}12x+(-13x)\end{align*}

6. \begin{align*}9z+(-9z)\end{align*}

7. \begin{align*}14a+(-3a)\end{align*}

8. \begin{align*}22y+(-33y)\end{align*}

9. \begin{align*}(-10d)+(-d)+2\end{align*}

10. \begin{align*}8x+(-4x)-5\end{align*}

11. \begin{align*}7y+(-3y)\end{align*}

12. \begin{align*}16x+(-22x)\end{align*}

13. \begin{align*}5a+(-a)+7a\end{align*}

Directions: Solve each real-world problem.

14. A plane is flying at an altitude that is 2, 500 feet above sea level. If the plane increases its altitude by 500 feet more, what will be its new altitude?

15. The temperature on a mountaintop at midnight was \begin{align*}-8^\circ F\end{align*}. By 3:00 A.M., the temperature had risen by \begin{align*}3^\circ F\end{align*}. What is the temperature at 3:00 A.M.?

### Notes/Highlights Having trouble? Report an issue.

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Term | Definition |
---|---|

Expression |
An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols. |

Integer |
The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3... |

Sum |
The sum is the result after two or more amounts have been added together. |

Variable |
A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n. |

Variable Expression |
A variable expression is a mathematical phrase that contains at least one variable or unknown quantity. |

### Image Attributions

Here you'll learn to evaluate and simplify variable expressions involving integer addition.

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