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# Numerical Expression Evaluation

## Use properties of equality and order of operations.

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Practice Numerical Expression Evaluation
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Evaluate Variable Expressions with Given Values

### [Figure1] License: CC BY-NC 3.0

Jenny is making cookies for the school bake sale. One dozen cookies will be sold for $3.50 and one dozen muffins will be sold for$4.50. To quickly calculate the amount of money raised, Jenny has written the following variable expression:

\begin{align*}3.50c+4.50m\end{align*}where \begin{align*}c=\end{align*}the number of dozens of cookies sold and \begin{align*}m=\end{align*}the number of dozens of muffins sold.

At the end of day, Jenny knows that 311 dozen cookies and 235 dozen muffins were sold. How can she use the variable expression to determine the total amount of money raised?

In this concept, you will learn to evaluate variable expressions with given values for the variables.

### Variable Expressions

Before you can evaluate a variable expression with a given value for the variable, you must be able to identify a variable expression.

A variable expression is a group of numbers and mathematical operations that contains one or more variables. A variable expression does not contain an equal sign.

A variable is a letter used to represent an unknown quantity.

The variable expression \begin{align*}3x-6+5y\end{align*} contains the variables \begin{align*}x\end{align*}and \begin{align*}y\end{align*}the constant six. A constant is a term in an expression that is simply a number.

When values for the variable or variables are given and substituted into the expression, the values should be written inside parenthesis. This will allow you to see what operations have to be performed to evaluate the expression.

Let’s look at an example of evaluating a variable expression when you have been given a value for the variable.

Evaluate the variable expression:

\begin{align*}4.2g+1.5 \ \text{when} \ g=8\end{align*}

To evaluate this variable expression you may have to review performing mathematical operations with decimals.

First, substitute \begin{align*}g=8\end{align*}into the variable expression.

\begin{align*}4.2(8)+1.5\end{align*}

Next, multiply: \begin{align*}4.2(8)=33.6\end{align*} to clear the parenthesis.  Write the new expression.

\begin{align*}33.6+1.5\end{align*}Then, add:

\begin{align*}33.6+1.5=35.1\end{align*}The answer is  35.1

Let’s look at an example of evaluating a variable expression that has more than one variable.

Evaluate the variable expression:

\begin{align*}5ab + 2a-7\end{align*} when \begin{align*}a=2\end{align*} and \begin{align*}b=4\end{align*}

First, substitute the values \begin{align*}a=2\end{align*} and \begin{align*}b=4\end{align*} into the variable expression.

\begin{align*}5(2 \times 4)+2(2)-7\end{align*}

Next, multiply: \begin{align*}(2 \times 4)=8\end{align*} and write the new expression:

\begin{align*}5(8)+2(2)-7\end{align*}

Next, multiply: \begin{align*}5(8)=40\end{align*} to clear the parenthesis. Write the new expression.

\begin{align*}40+2(2)-7\end{align*}

Next, multiply: \begin{align*}2(2)=40\end{align*} to clear the parenthesis. Write the new expression.

\begin{align*}40+4-7\end{align*}

Next, add: \begin{align*}40+4=44\end{align*} and write the new expression.

\begin{align*}44-7\end{align*}Then, subtract:

\begin{align*}44-7=37\end{align*}The answer is 37.

### Examples

#### Example 1

Earlier, you were given a problem about Jenny and her bake sale.

Jenny wrote a variable expression \begin{align*}3.50c+4.50m\end{align*} where \begin{align*}c\end{align*} represents the number of dozens of cookies and \begin{align*}m\end{align*}represents the number of dozens of muffins.

There were 311 dozen cookies and 235 dozen muffins sold.

First, substitute the values \begin{align*}c=311\end{align*}and \begin{align*}m=235\end{align*} into the variable expression.\begin{align*}3.50(311)+4.50(2.35)\end{align*}

Next, clear the parentheses by multiplying.

\begin{align*}3.50(311)=1088.50 \ \text{and} \ 4.50(235)=1057.50 \end{align*}

Then, add: \begin{align*}& \quad \ \ \1088.50 \\ & \underline{\;\;\; + \ 1057.50} \\ & \ \ \ \ \ \ \2146.00\end{align*}

#### Example 2

Evaluate the following variable expression:

\begin{align*}5a+2b-17\end{align*} when \begin{align*}a=3\end{align*} and \begin{align*}b=4\end{align*}

First, substitute the values \begin{align*}a=3\end{align*} and \begin{align*}b=4\end{align*}into the expression.

\begin{align*}5(3)+2(4)-17\end{align*}
Next, multiply:\begin{align*}5(3)=15\end{align*} to clear the parenthesis.

\begin{align*}15+2(4)-17\end{align*}
Next, multiply: \begin{align*}2(4)=8\end{align*} to clear the parenthesis.

\begin{align*}15+8-17\end{align*}Then, add and subtract, in order from left to right:

\begin{align*}15+8=23-17=6\end{align*}The answer is 6.

#### Example 3

Evaluate the following variable expression:

\begin{align*}6xy+2x-7\end{align*} when \begin{align*}x=4\end{align*} and \begin{align*}y=5\end{align*}

First, substitute the values \begin{align*}x=4\end{align*} and \begin{align*}y=5\end{align*} into the variable expression.

\begin{align*}6(4)(5)+2(4)-7\end{align*}
Next, multiply:\begin{align*}6(4)=24\end{align*} to clear the parenthesis.

\begin{align*}24(5)+2(4)-7\end{align*}
Next, multiply:\begin{align*}24(5)=120\end{align*} to clear the parenthesis. \begin{align*}120+2(4)-7\end{align*}Next, multiply: \begin{align*}2(4)=8\end{align*} to clear the parenthesis.

\begin{align*}120+8-7\end{align*}Then, add and subtract, in order from left to right.

\begin{align*}120+8=128-7=121\end{align*}The answer is 121.

#### Example 4

Evaluate the following variable expression:

\begin{align*}9x+18y+5\end{align*} when \begin{align*}x=-6\end{align*} and \begin{align*}y=2\end{align*}

First, substitute the values \begin{align*}x=-6\end{align*} and \begin{align*}y=2\end{align*} into the variable expression.

\begin{align*}9(-6)+18(2)+5 \end{align*}
Next, multiply:\begin{align*}9(-6)=-54\end{align*} to clear the parenthesis.

\begin{align*}-54+18(2)+5\end{align*}

Next, multiply:\begin{align*}18(2)=36\end{align*} to clear the parenthesis.

\begin{align*}-54+36+5\end{align*}

Then, add and subtract, in order from left to right.

\begin{align*}-54+36=-18+5=-13\end{align*}

### Review

Evaluate each variable expression using the given values for each variable.

1. \begin{align*}6a+7\end{align*} when \begin{align*}a\end{align*} is 8
2. \begin{align*}9x−y\end{align*} when \begin{align*}x\end{align*} is 2 and \begin{align*}y\end{align*} is 4
3. \begin{align*}5a+b^2\end{align*} when \begin{align*}a\end{align*} is 12 and \begin{align*}b\end{align*} is 4
4. \begin{align*}\left(\frac{8}{x}\right)+2\end{align*} when \begin{align*}x\end{align*} is 4
5. \begin{align*} 6x + 2.5\end{align*} when \begin{align*}x\end{align*} is 2
6. \begin{align*}y^2 + 4\end{align*} when \begin{align*}y\end{align*} is 9
7. \begin{align*}7x + 2y\end{align*} when \begin{align*}x\end{align*} is 3 and y is 5
8. \begin{align*}9xy + x^2\end{align*} when \begin{align*}x\end{align*} is 4 and \begin{align*}y\end{align*} is 2
9. \begin{align*}3ab + b^3\end{align*} when \begin{align*}a\end{align*} is 9 and \begin{align*}b\end{align*} is 2
10. \begin{align*}16xy^2 + 14\end{align*} when  \begin{align*}x\end{align*} is 3 and \begin{align*}y\end{align*} is 4
11. \begin{align*}6xy + 4x\end{align*} when \begin{align*}x\end{align*} is 2 and \begin{align*}y\end{align*} is 7
12. \begin{align*}16y + 8xy\end{align*} when \begin{align*}x\end{align*} is 3 and \begin{align*}y\end{align*} is 4
13. \begin{align*}3x^2 + 24y\end{align*} when \begin{align*}x\end{align*} is 3 and \begin{align*}y\end{align*} is 4
14. \begin{align*}–xy + 8xy\end{align*} when \begin{align*}x\end{align*} is 3 and \begin{align*}y\end{align*} is 4
15. \begin{align*}22x + 4y − 3xy\end{align*} when \begin{align*}x\end{align*} is 3 and \begin{align*}y\end{align*} is 4

To see the Review answers, open this PDF file and look for section 1.8.

### Vocabulary Language: English

constant

A constant is a value that does not change. In Algebra, this is a number such as 3, 12, 342, etc., as opposed to a variable such as x, y or a.

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.

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.