# 1.7.3: Simplify Variable Expressions Involving Multiple Operations

- Page ID
- 4318

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Simplify Variable Expressions Involving Multiple Operations

A farmer, Kelly, has two lovely plots of rectangular land to grow some vegetables. Her friend will help her plant in the spring. One plot of land is 8a by 6a and the other plot of land is 3a by 4a. The two plots of land are going to be combined so they can grow more vegetables. How can Kelly find the total area for both plots of farmable land?

In this concept, you will learn to **simplify** variable expressions involving multiple operations.

**Simplifying Variable Expressions Involving Multiple Operations**

Sometimes, you may need to simplify algebraic expressions that involve more than one operation. Use what you know about simplifying sums, differences, products, or quotients of algebraic expressions to help you do this.

When evaluating expressions, it is important to keep in mind the **order of operations**, which is

- First, do the computation inside parentheses.
- Second, evaluate any exponents.
- Third, multiply and divide in order from left to right.
- Finally, add and subtract in order from left to right.

Now let’s look at an example.

Simplify this **expression** 7n+8n⋅3

First, simplify according to the order of operations. According to the order of operations, you should multiply first.

7n+8n⋅3=7n+24n.

Next, add like terms.

7n+24n=31n

The answer is 31n.

Here is another example.

Simplify the expression 10p−7p+8p÷2p.

First, follow the order of operations and rewrite the division as a fraction.

8p/2p

Next, simplify the fraction, assuming p is not equal to zero.

8p/2p=8/2×p/p=4×1=4

Next, rewrite the equation.

10p−7p+(8p/2p)=10p−7p+4

Simplify, by combing like terms.

10p−7p+4=3p+4

The answer is 3p+4.

**Examples**

Example \(\PageIndex{1}\)

Earlier, you were given a problem about Kelly and her two plots of land.

One is 8a by 6a and the other is 3a by 4a.

The plots of land need to be combined to find the total area of farmable land.

**Solution**

First, consider the equation for the area of a rectangle.

Area of a rectangle=length×width

Next, calculate the area of the first rectangular plot of land.

8a×6a=48a^{2}

Then, calculate the area of the second rectangular plot of land.

3a×4a=12a^{2}

Next, write and expression for the total area of the two plots of land.

48a^{2}+12a^{2}

Finally combine like terms.

48a^{2}+12a^{2}=60a^{2}

The answer is 60a^{2}.

Example \(\PageIndex{1}\)

Samera has twice as many pets as Amit has. Kyra has 4 times as many pets as Amit has. Let a represent the number of pets Amit has.

- Write an expression to represent the number of pets Samera has.
- Write an expression to represent the number of pets Kyra has.
- Write an expression to represent the number of pets Samera and Kyra have all together.

**Solution**

First, answer part 1.

Samera has twice as many pets as Amit. Since Amit has a pets, Samera has 2a pets.

Next, answer part 2.

Kyra has 4 times as many pets as Amit has. Since Amit has a pets, Kyra has 4a pets.

Finally, answer part 3.

To find the number of pets Samera and Kyra have “all together,” write an addition expression.

2a+4a

Finally, combine like terms.

2a+4a=6a

The answer is 6a.

**Simplify each expression.**

Example \(\PageIndex{1}\)

4a+9a−7

**Solution**

Combine the like terms 4a and 9a.

The answer is 13a−7.

Example \(\PageIndex{1}\)

(14x/2)+9x

**Solution**

First, follow the order of operations and do the division first.

(14x/2)+9x=7x+9x

Next, combine like terms.

7x+9x=16x

The answer is 16x.

Example \(\PageIndex{1}\)

6b−2b+5b−8

**Solution**

Follow the order of operations and perform the addition and subtraction from left to right.

6b−2b+5b−8=9b−8

The answer is 9b−8.

### Review

Simplify each expression involving multiple operations.

- 6a+4a−2b
- 16b−4b⋅2
- 22a÷2+14a
- 19x−5x⋅2
- 16y−12y÷2
- 16a−4a−12b
- 26a+14a+12b+2b
- 36a+4a−2b+5b
- 18a+4a+12y
- 46a+34a−12b+14b
- 16y+4y−2x
- 6x+4x+2x+4y−19z
- 26y−12y÷2
- 36y−12y÷12
- 46y+12y÷2

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.6.

### Vocabulary

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

Difference |
The result of a subtraction operation is called a difference. |

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. |

Product |
The product is the result after two amounts have been multiplied. |

Quotient |
The quotient is the result after two amounts have been divided. |

Simplify |
To simplify means to rewrite an expression to make it as "simple" as possible. You can simplify by removing parentheses, combining like terms, or reducing fractions. |

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

### Additional Resources

Video:

Practice: **Simplify Variable Expressions Involving Multiple Operations**