# Single Variable Expressions

## Use symbols and operations to understand and define variables.

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Single Variable Expressions

Shelly is making bracelets to sell at her town's market this summer. She spent $150 on supplies and will make$4 for every bracelet she sells. Her profit for selling \begin{align*}b\end{align*} bracelets is given by the expression \begin{align*}4b-150.\end{align*} How can Shelly calculate her profit if she sells 50 bracelets this summer?

In this concept, you will learn how to evaluate single variable expressions.

### Evaluating Single Variable Expressions

An expression is a mathematical phrase that contains numbers and operations.

Here are some examples of expressions:

•   \begin{align*}3+10-5\end{align*}
• \begin{align*}-15+7-1\end{align*}
• \begin{align*}5^2-1\end{align*}
•  \begin{align*}15(3 + 4)+2\end{align*}

A variable is a symbol or letter (such as \begin{align*}x,m,R,y,P\end{align*}, or \begin{align*}a\end{align*}) that is used to represent a quantity that might change in value. A variable expression is an expression that includes variables. Another name for a variable expression is an algebraic expression.

Here are some examples of variable expressions:

• \begin{align*}3x+10\end{align*}
• \begin{align*}10r\end{align*}
• \begin{align*}b^3+2\end{align*}
• \begin{align*}mx-3\end{align*}

A single variable expression is a variable expression with just one variable in it.

You can use a variable expression to describe a real world situation where one or more quantities has an unknown value or can change in value.

To evaluate a variable expression means to find the value of the expression for a given value of the variable. To evaluate, substitute the given value for the variable in the expression and simplify using the order of operations. To follow the order of operations, you always need to do any multiplication/division first before any addition/subtraction.

Here is an example.

Evaluate the expression \begin{align*}10k-44\end{align*} for \begin{align*}k=12\end{align*}.

First, remember that when you see a number next to a letter, like “\begin{align*}10k\end{align*}”, it means to multiply.

Next, substitute 12 in for the letter \begin{align*}k\end{align*} in the expression.

\begin{align*}10(12)-44\end{align*}

Notice that you can put parentheses around the 12 to keep it separate from the number 10.

Now, simplify the expression using the order of operations. You will need to multiply first and then subtract.

\begin{align*}\begin{array}{rcl} 10(12)-44 &=& 120-44 \\ &=& 76 \end{array}\end{align*}

Here is another example that involves division.

Evaluate the expression \begin{align*}\frac{x}{3}+2\end{align*} for \begin{align*}x=24\end{align*}.

First, remember that a fraction bar is like a division sign. \begin{align*}\frac{x}{3}\end{align*} is the same as \begin{align*}x \div 3\end{align*}.

Next, substitute 24 in for the letter \begin{align*}x\end{align*} in the expression.

\begin{align*}\frac{24}{3}+2\end{align*}

Now, simplify the expression using the order of operations. You will need to divide first and then add.

\begin{align*}\begin{array}{rcl} \frac{24}{3}+2 &=&8+2\\ &=& 10 \end{array} \end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Shelly and her bracelet business.

Shelly is selling bracelets this summer and her profit for selling \begin{align*}b\end{align*} bracelets is given by the expression \begin{align*}4b-150.\end{align*} Shelly wants to calculate what her profit will be if she sells 50 bracelets.

To calculate her profit from selling 50 bracelets, Shelly needs to evaluate the expression \begin{align*}4b-150\end{align*} for \begin{align*}b=50.\end{align*}

First, substitute 50 in for the letter \begin{align*}b\end{align*} in the expression.

\begin{align*}4(50)-150\end{align*}

Now, simplify the expression using the order of operations. You will need to multiply first and then subtract.

\begin{align*}\begin{array}{rcl} 4(50)-150 &=& 200-150\\ &=& 50 \end{array}\end{align*}

Shelly's profit from selling 50 bracelets would be \$50.

#### Example 2

Evaluate \begin{align*}\frac{x}{7}-5\end{align*} if \begin{align*}x\end{align*} is 49.

First, substitute 49 in for the letter \begin{align*}x\end{align*} in the expression.

\begin{align*}\frac{49}{7}-5\end{align*}

Now, simplify the expression using the order of operations. You will need to divide first and then subtract.

\begin{align*}\begin{array}{rcl} \frac{49}{7}-5 &=& 7-5 \\ &=& 2 \end{array}\end{align*}

#### Example 3

Evaluate \begin{align*}4x-9\end{align*} if \begin{align*}x\end{align*} is 20.

First, substitute 20 in for the letter \begin{align*}x\end{align*} in the expression.

\begin{align*}4(20)-9\end{align*}

Now, simplify the expression using the order of operations. You will need to multiply first and then subtract.

\begin{align*}\begin{array}{rcl} 4(20)-9 &=& 80-9 \\ &=& 71 \end{array}\end{align*}

#### Example 4

Evaluate \begin{align*}5y+6\end{align*} if \begin{align*}y\end{align*} is 9.

First, substitute 9 in for the letter \begin{align*}y\end{align*} in the expression.

\begin{align*}5(9)+6\end{align*}

Now, simplify the expression using the order of operations. You will need to multiply first and then add.

\begin{align*}\begin{array}{rcl} 5(9)+6 &=& 45+6\\ &=& 51 \end{array}\end{align*}

#### Example 5

Evaluate \begin{align*}\frac{a}{4}-8\end{align*} if \begin{align*}a\end{align*} is 36.

First, substitute 36 in for the letter \begin{align*}a\end{align*} in the expression.

\begin{align*}\frac{36}{4}-8\end{align*}

Now, simplify the expression using the order of operations. You will need to divide first and then subtract.

\begin{align*}\begin{array}{rcl} \frac{36}{4}-8 &=& 9-8\\ &=& 1 \end{array}\end{align*}

### Review

Evaluate each expression if the given value of \begin{align*}r\end{align*} is 9.

1. \begin{align*}\frac{r}{3}\end{align*}
2. \begin{align*}63 - r\end{align*}
3. \begin{align*}11r\end{align*}
4. \begin{align*}2r + 7\end{align*}
5. \begin{align*}3r + r\end{align*}
6. \begin{align*}4r - 2r\end{align*}
7. \begin{align*}r + 5r\end{align*}
8. \begin{align*}12r - 1\end{align*}

Evaluate each expression for \begin{align*}h=12\end{align*}.

1. \begin{align*}70 - 3h\end{align*}
2. \begin{align*}6h + 6\end{align*}
3. \begin{align*}4h - 9\end{align*}
4. \begin{align*}11 + \frac{h}{4}\end{align*}
5. \begin{align*}3h + h\end{align*}
6. \begin{align*}2h + 5h\end{align*}
7. \begin{align*}6h - 2h\end{align*}

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### Vocabulary Language: English

TermDefinition
Algebraic Expression An expression that has numbers, operations and variables, but no equals sign.
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.