Remember Kelly and her pre-trip meeting from the Single Variable Expressions Concept? Have you ever been hiking on difficult terrain?

Kelly figured out how many miles per hour her group would be able to hike on difficult terrain. However, she also had other thoughts about flat terrain. Look at this information.

“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 \begin{align*}\frac{1}{2}\end{align*} miles to 2 miles per hour.”

This is what the group leader told Kelly and the other hikers. Kelly has figured out the difficult hiking and now she has another idea of a problem to solve. Kelly wants to figure out the number of miles possible between the range of \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.

**Pay close attention during this Concept and you will know how to use a variable expression to help Kelly figure things out.**

### Guidance

Previously we worked on how to evaluate algebraic expressions with a single variable. Well, algebraic expressions can have more than one variable. Look at the following situations with multi-variable expressions.

\begin{align*}& xy + 4x\\ & mx + b\\ &25r + (x - 7)\\ & x + y + z\end{align*}

When we know the value of the variables, we can evaluate multi-variable expressions the same way we evaluated single-variable expressions, by substituting the value for the variables in the expression and solving from left to right.

Evaluate \begin{align*}xy + x\end{align*} *if* \begin{align*}x = 2\end{align*} *and* \begin{align*}y = 4.\end{align*}

In this case we are only given one possible value for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. We know that \begin{align*}x = 2\end{align*} and \begin{align*}y = 4\end{align*}, we can evaluate the expression using the given values.

**First, we can rewrite the expression by substituting the given values into the equation.**

\begin{align*}& xy + x\\ & (2)(4) + 2\end{align*}

We used the parentheses here to show multiplication. When two variables are next to each other it means multiplication. Here we used the parentheses because we needed to show multiplication between 2 and 4.

Now we can multiply first.

\begin{align*}2 \times 4 = 8\end{align*}

Next, we add two.

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

**Our answer is 10.**

Let’s look at another one where a given value is a fraction.

Evaluate \begin{align*}mx + 3m\end{align*} *if* \begin{align*}x = \frac{2}{3}\end{align*} *and* \begin{align*}m = 9\end{align*}.

First, we substitute the given values into the expression.

\begin{align*}9 \left(\frac{2}{3}\right)+3(9)\end{align*}

Do you remember how to multiply a whole number and a fraction?

**To multiply a whole number and a fraction you must first make the whole number a fraction over one.**

9 becomes \begin{align*}\frac{9}{1}\end{align*}

Now you multiply numerator \begin{align*}\times\end{align*} numerator and denominator \begin{align*}\times\end{align*} denominator.

\begin{align*}\frac{9}{1} \cdot \frac{2}{3}+ 3(9)\end{align*}

Next, we multiply the two fractions and simplify.

\begin{align*}\frac{18}{3} = 6\end{align*}

Now we can substitute the 6 back into the expression.

\begin{align*}6 + 3(9)\end{align*}

Next we multiply.

\begin{align*}3(9) = 27\end{align*}

**Finally, we add the remaining numbers.**

\begin{align*}& 6 + 27\\ & 33\end{align*}

**Our answer is 33.**

Evaluate the following expressions using the given values.

#### Example A

\begin{align*}ab+7\end{align*} ** when** \begin{align*}a\end{align*}

*is***9**

**\begin{align*}b\end{align*}**

*and*

*is***8**

**Solution: 79**

#### Example B

\begin{align*}xy+zx\end{align*} ** when** \begin{align*}x\end{align*}

*is***2,**\begin{align*}y\end{align*}

*is***5**

**\begin{align*}z\end{align*}**

*and*

*is***7**

**Solution: 24**

#### Example C

\begin{align*}xy+x\end{align*} ** when** \begin{align*}x\end{align*}

**\begin{align*}\frac{1}{4}\end{align*},**

*is***\begin{align*}y\end{align*}**

*and***\begin{align*}\frac{4}{5}\end{align*}**

*is*
**Solution: \begin{align*}\frac{9}{20}\end{align*}**

Now back to Kelly. Once she has figured out that the group will probably go three miles per hour on difficult terrain, she moved on to other terrains.

**Next, Kelly looks at medium terrain. She substitutes 1 mile into the expression for \begin{align*}x\end{align*}.**

\begin{align*}6(1)=6\end{align*} *miles on medium terrain*

**Next, Kelly looks at flat terrain. She substitutes 2 miles in for \begin{align*}x\end{align*}.**

\begin{align*}6(2) = 12\end{align*} *miles on flat terrain*

**Kelly starts to think about this. It would be unlikely for the group to travel on all one type of terrain per day. So she writes this expression to show half the day on flat terrain and half the day on difficult terrain.**

\begin{align*}x=\end{align*} ** flat terrain** \begin{align*}=2\end{align*}

*miles*
\begin{align*}y=\end{align*} ** difficult terrain** \begin{align*}=\frac{1}{2}\end{align*}

*mile*\begin{align*}& 3x+3y\\ & 3(2)+ 3\left(\frac{1}{2}\right)\\ & 6+1 \frac{1}{2} \ miles=7 \frac{1}{2} \ miles\end{align*}

**Kelly looks at this figure. While she is estimating travel time and distance per miles, she figures that the group may cover a little less than 7 and one-half miles or a little more, but it is probably a good middle estimate for distance covered per 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.

\begin{align*}xy+xy\end{align*} ** when** \begin{align*}x\end{align*}

**\begin{align*}\frac{1}{2}\end{align*},**

*is***\begin{align*}y\end{align*}**

*and***\begin{align*}\frac{2}{3}\end{align*}**

*is*
**Answer**

To evaluate this expression, we can substitute the given values into the expression for the variables.

\begin{align*}\frac{1}{2}(\frac{2}{3}) + \frac{1}{2}(\frac{2}{3})\end{align*}

\begin{align*}\frac{2}{6} + \frac{2}{6} = \frac{4}{6} = \frac{2}{3}\end{align*}

**This is our answer.**

### Video Review

This is a James Sousa video on evaluating algebraic expressions.

### Practice

Directions: Evaluate each multi-variable expression *if* \begin{align*}x = 3\end{align*} *and* \begin{align*}y = 4\end{align*}

1. Evaluate \begin{align*}xy + 2y\end{align*}

2. Evaluate \begin{align*}3y + 2y\end{align*}

3. Evaluate \begin{align*}3y + 9x\end{align*}

4. Evaluate \begin{align*}xy + 3xy\end{align*}

5. Evaluate \begin{align*}2xy + 9xy\end{align*}

Directions: Evaluate each multi-variable expression *if* \begin{align*}x=10\end{align*} *and* \begin{align*}y=5\end{align*}

6. Evaluate \begin{align*}5x+xy\end{align*}

7. Evaluate \begin{align*}3x+2y\end{align*}

8. Evaluate \begin{align*}2x+3y\end{align*}

9. Evaluate \begin{align*}4x+3y+x\end{align*}

10. Evaluate \begin{align*}5x+3y+2x\end{align*}

Directions: Evaluate each multi-variable expression *if* \begin{align*}y = 2\end{align*} *and* \begin{align*}z = 4\end{align*}

11. Evaluate \begin{align*}2y + 3y - 2z\end{align*}

12. Evaluate \begin{align*}5y + 3y - 3z\end{align*}

13. Evaluate \begin{align*}2y + 5y - 2z\end{align*}

14. Evaluate \begin{align*}4y + 3y - 2z\end{align*}

15. Evaluate \begin{align*}6z-2(z+x)\end{align*} *if* \begin{align*}x\end{align*} *is* 3 *and* \begin{align*}z\end{align*} *is* 4

16. Evaluate \begin{align*}8a + 3b - 2c\end{align*} *if* \begin{align*}a\end{align*} *is* 5, \begin{align*}b\end{align*} *is* 4 *and* \begin{align*}c\end{align*} *is* 3