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# Single Variable Expressions

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

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Single Variable Expressions

After waiting for what seemed like an eternity, the day of Kelly’s pre-trip meeting finally arrived. She had been sent a list of recommended supplies that she needed to bring to the meeting so that the leaders could check her gear. Kelly gathered her things and put everything in the family van. Then her Dad went with her to the pre-trip meeting.

“I am so excited,” she said to her Dad on the way over.

The pre-trip meeting was great. Kelly met several other hikers. They would be organized into hiking groups later, so today they all set together as one big crew. There were six leaders and thirty students with an even mix of boys and girls. While parents had coffee and tea, the leaders talked about trail safety, personal safety and about skills that they would learn.

The leaders said that the students would be hiking for about 6 hours per day with breaks.

“Wow,” thought Kelly “This is going to be something.”

Each day the group would have a goal of where they would start and where they would stop.

“The amount of distance traveled depends on the group,” said Scott one of the leaders. “If your group works well together, they you can cover about 1 – 2 miles per hour on flat terrain. If you are climbing higher ascents, then you may only cover 12\begin{align*}\frac{1}{2}\end{align*} miles to 2 miles per hour.”

On the way home, Kelly started thinking about those variables. It seemed to her like there could be a range of distances covered even if they were hiking 6 hours per day. Kelly took out a notebook and made some notes.

6 hours of hiking per day

x\begin{align*}x\end{align*} number of miles hiked per hour

Kelly wants to figure out the hardest hiking first. She decides to calculate the time it would take if she and her group were only able to cover 12\begin{align*}\frac{1}{2}\end{align*} miles per hour. Pay attention and you will be able to help Kelly with this problem at the end of the Concept.

### Guidance

An algebraic expression is a mathematical phrase involving letters, numbers, and operation symbols.

A variable can be any letter, such as x, m, R, y, P, a\begin{align*}x, \ m, \ R, \ y, \ P, \ a\end{align*}, and others, that we use in an expression.

The variable represents possible values of a quantity.

Take a look at the following variable expressions.

3x+1010rb3+2mx3

Variable expressions are used to describe real-world situations when we don’t know a value or a quantity. Sometimes a value is dependent on a changing variable.

When we work with a variable expression, we can say that we evaluate the expression. To evaluate means to “find the value of”. We find the value of an expression.

Why don’t we “solve” expressions?

To solve something in math means that it has to equal a value. An expression does not have an equal sign. Therefore we can’t solve it. Also, a variable in an expression can have several different values. Whereas in an equation, the variable has one value that makes the equation true.

Evaluate the expression 10k44\begin{align*}10k - 44\end{align*} if k=12\begin{align*}k = 12\end{align*}.

A number next to a letter means that we multiply. This is a good time to blast back in math and think about how we show multiplication and division besides using an x\begin{align*}x\end{align*} or a ÷\begin{align*}\div\end{align*} sign.

10k44\begin{align*}10k - 44\end{align*} if k=12\begin{align*}k = 12\end{align*}

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

10(12) – 44

Now we multiply 10 and 12, then we subtract 44.

1204476

Sometimes, you will also have an expression to evaluate that uses division.

Evaluate the expression x3+2\begin{align*}\frac{x}{3}+ 2\end{align*} if x\begin{align*}x\end{align*} is 24

First, we substitute the 24 in for x\begin{align*}x\end{align*}.

243+2\begin{align*}\frac{24}{3}+2\end{align*}

Next, we divide twenty-four by three.

24÷3=8\begin{align*}24 \div 3 = 8\end{align*}

8+2=10\begin{align*}8 + 2 = 10\end{align*}

Now it is your turn. Evaluate the following single-variable expressions.

#### Example A

4x9\begin{align*}4x-9\end{align*} if x\begin{align*}x\end{align*} is 20

Solution: 71

#### Example B

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

Solution: 51

#### Example C

a48\begin{align*}\frac{a}{4}-8\end{align*} if a\begin{align*}a\end{align*} is 36

Solution: 1

Now back to Kelly and her hiking calculations. Here is the original problem once again.

After waiting for what seemed like an eternity, the day of Kelly’s pre-trip meeting finally arrived. She had been sent a list of recommended supplies that she needed to bring to the meeting so that the leaders could check her gear. Kelly gathered her things and put everything in the family van. Then her Dad went with her to the pre-trip meeting.

“I am so excited,” she said to her Dad on the way over.

The pre-trip meeting was great. Kelly met several other hikers. They would be organized into hiking groups later, so today they all set together as one big crew. There were six leaders and thirty students with an even mix of boys and girls. While parents had coffee and tea, the leaders talked about trail safety, personal safety and about skills that they would learn.

The leaders said that the students would be hiking for about 6 hours per day with breaks.

“Wow,” thought Kelly “This is going to be something.”

Each day the group would have a goal of where they would start and where they would stop.

“The amount of distance traveled depends on the group,” said Scott one of the leaders. “If your group works well together, they you can cover about 1–2 miles per hour on flat terrain. If you are climbing higher ascents, then you may only cover 12\begin{align*}\frac{1}{2}\end{align*} miles to 2 miles per hour.”

On the way home, Kelly started thinking about those variables. It seemed to her like there could be a range of distances covered even if they were hiking 6 hours per day. Kelly took out a notebook and made some notes.

6 hours of hiking per day

x\begin{align*}x\end{align*} number of miles hiked per hour

Kelly wants to figure out the number of miles possible between the range of 12\begin{align*}\frac{1}{2}\end{align*} mile per hour and 2 miles per hour. She is sure that there is a way to do it using a variable and the six hours that the group will hike.

First, let’s write an expression to show the number of hours hiking times the possible number of miles. Since the number of miles can vary, we use the variable x\begin{align*}x\end{align*} for this distance.

6x\begin{align*}6x\end{align*}

First, Kelly substitutes 12\begin{align*}\frac{1}{2}\end{align*} for x\begin{align*}x\end{align*} to figure out about how many miles they will cover if all six hours is hiking steep terrain.

\begin{align*}6\left(\frac{1}{2}\right)= \frac{6}{2}=3 \ miles\end{align*}

Wow, on steep terrain the group will probably only cover about 3 miles in the entire day.

### Vocabulary

Algebraic Expression
an expression that contains numbers, variables and operations.
Variable
a letter used to represent an unknown quantity.
Variable Expression
an algebraic expression that contains one or more variables.

### Guided Practice

Here is one for you to try on your own.

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

To evaluate this expression, we substitute 49 into the expression for x.

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

Next, we evaluate the expression.

\begin{align*}7 - 5 = 2\end{align*}

### Practice

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

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*}

Directions: Evaluate each expression when \begin{align*}h=12\end{align*}.

9. \begin{align*}70 - 3h\end{align*}

10. \begin{align*}6h + 6\end{align*}

11. \begin{align*}4h - 9\end{align*}

12. \begin{align*}11 + \frac{h}{4}\end{align*}

13. \begin{align*}3h+h\end{align*}

14. \begin{align*}2h+5h\end{align*}

15. \begin{align*}6h-2h\end{align*}

### Vocabulary Language: English

Algebraic Expression

Algebraic Expression

An expression that has numbers, operations and variables, but no equals sign.
Variable

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.