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Simplify Variable Expressions Involving Multiple Operations

Perform the order of operations to simplify variable expressions.

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Simplify Variable Expressions Involving Multiple Operations
License: CC BY-NC 3.0

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 \begin{align*}8a\end{align*}8a by \begin{align*}6a\end{align*}6a and the other plot of land is \begin{align*}3a\end{align*}3a by \begin{align*}4a\end{align*}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 \begin{align*}7n+8n \cdot 3\end{align*}7n+8n3

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

\begin{align*}7n + 8n \cdot 3 = 7n + 24n.\end{align*}7n+8n3=7n+24n.

Next, add like terms.

\begin{align*}7n+24n=31n\end{align*}7n+24n=31n

The answer is \begin{align*}31n\end{align*}31n.

Here is another example.

Simplify the expression \begin{align*}10p-7p+8p \div 2p\end{align*}10p7p+8p÷2p.

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

\begin{align*}\frac{8p}{2p}\end{align*}8p2p

Next, simplify the fraction, assuming \begin{align*}p\end{align*}p is not equal to zero.

\begin{align*}\frac{8p}{2p}= \frac{8}{2} \times \frac{p}{p}=4 \times 1=4 \end{align*}8p2p=82×pp=4×1=4

Next, rewrite the equation.

\begin{align*}10p-7p+ \frac{8p}{2p}=10p-7p+4\end{align*}10p7p+8p2p=10p7p+4

Simplify, by combing like terms.

\begin{align*}10p-7p+4=3p+4\end{align*}10p7p+4=3p+4

The answer is \begin{align*}3p+4\end{align*}3p+4.

Examples

Example 1

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

One is \begin{align*}8a\end{align*}8a by \begin{align*}6a\end{align*} and the other is \begin{align*}3a\end{align*} by \begin{align*}4a\end{align*}.

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

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

\begin{align*}\text{Area of a rectangle} = \text{length} \times \text{width}\end{align*}

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

\begin{align*}8a \times 6a=48a^2\end{align*}

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

\begin{align*}3a \times 4a=12a^2\end{align*}

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

\begin{align*}48a^2+12a^2\end{align*}

Finally combine like terms.

\begin{align*}48a^2+12a^2= 60a^2\end{align*}

The answer is \begin{align*}60a^2\end{align*}.

Example 2

Samera has twice as many pets as Amit has. Kyra has 4 times as many pets as Amit has. Let \begin{align*}a\end{align*} represent the number of pets Amit has.

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

First, answer part a.

Samera has twice as many pets as Amit. Since Amit has a pets, Samera has \begin{align*}2a\end{align*} pets.

Next, answer part b.

Kyra has 4 times as many pets as Amit has. Since Amit has a pets, Kyra has \begin{align*}4a\end{align*} pets.

Finally, answer part c.

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

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

Finally, combine like terms.

\begin{align*}2a+4a=6a\end{align*}

The answer is \begin{align*}6a\end{align*}.

Simplify each expression.

Example 3

\begin{align*}4a + 9a-7\end{align*}

Combine the like terms \begin{align*}4a\end{align*} and \begin{align*}9a\end{align*}.

The answer is \begin{align*}13a-7\end{align*}.

Example 4

\begin{align*}\frac{14x}{2}+9x\end{align*}

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

\begin{align*}\frac{14x}{2}+9x=7x+9x\end{align*}

Next, combine like terms.

\begin{align*}7x+9x=16x\end{align*}

The answer is \begin{align*}16x\end{align*}.

Example 5

\begin{align*}6b-2b + 5b-8\end{align*}

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

\begin{align*}6b-2b+5b-8=9b-8\end{align*}

The answer is \begin{align*}9b-8\end{align*}.

Review

Simplify each expression involving multiple operations.

  1. \begin{align*}6a+4a-2b\end{align*}
  2. \begin{align*}16b-4b \cdot 2\end{align*} 
  3. \begin{align*}22a \div 2 + 14a\end{align*} 
  4. \begin{align*}19x-5x \cdot 2\end{align*} 
  5. \begin{align*}16y-12y \div 2\end{align*} 
  6. \begin{align*}16a-4a-12b\end{align*} 
  7. \begin{align*}26a + 14a + 12b + 2b\end{align*}
  8. \begin{align*}36a + 4a-2b + 5b\end{align*} 
  9. \begin{align*}18a + 4a + 12y\end{align*} 
  10. \begin{align*}46a + 34a-12b + 14b\end{align*}
  11. \begin{align*}16y + 4y-2x\end{align*} 
  12. \begin{align*}6x + 4x + 2x + 4y-19z\end{align*} 
  13. \begin{align*}26y-12y \div 2\end{align*} 
  14. \begin{align*}36y-12y \div 12\end{align*} 
  15. \begin{align*}46y + 12y \div 2\end{align*}

Review (Answers)

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

Resources

 

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Vocabulary

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.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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