1.2: Using Expressions
Introduction
The Pre-trip Meeting
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 sat 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 that they would learn.
The leaders said that the students would be hiking 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 distance traveled depends on the group,” said Scott, one of the leaders. “If your group works well together, then 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*} mile 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
\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 \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. As Kelly starts figuring things out, it is time for you to do the same. This lesson is all about expressions. Using expressions is very useful and you will see how useful they are for Kelly at the end of this lesson.
What You Will Learn
In this lesson you will learn the following skills.
- Evaluate single-variable expressions with given values for the variable.
- Evaluate multi-variable expressions with given values for the variables.
- Use given expressions to analyze and solve real-world problems.
- Write variable expressions to represent and solve real-world problems.
Teaching Time
I. Evaluate Single-Variable Expressions with Given Values for the Variable
An algebraic expression is a mathematical phrase involving letters, numbers, and operation symbols.
A variable can be any letter, such as \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 examples of variable expressions
\begin{align*}& 3x + 10\\ & 10r\\ & b^3 + 2\\ & mx - 3\end{align*}
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.
Example
Evaluate the expression \begin{align*}10k - 44\end{align*} if \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 \begin{align*}x\end{align*} or a \begin{align*}\div\end{align*} sign.
Back to Our Example
\begin{align*}10k - 44\end{align*} if \begin{align*}k = 12\end{align*}
First, we substitute the 12 for the letter \begin{align*}k\end{align*}.
10(12) – 44
Now we multiply 10 and 12 (following the order of operations), then we subtract 44.
\begin{align*}120 - 44\!\\ 76\end{align*}
Our answer is 76.
Sometimes, you will also have an expression to evaluate that uses division.
Example
Evaluate the expression \begin{align*}\frac{x}{3}+ 2\end{align*} if \begin{align*}x\end{align*} is 24
First, we substitute the 24 for \begin{align*}x\end{align*}.
\begin{align*}\frac{24}{3}+2\end{align*}
Next, we divide twenty-four by three.
\begin{align*}24 \div 3 = 8\end{align*}
Now we add 2.
\begin{align*}8 + 2 = 10\end{align*}
Our answer is 10.
1E. Lesson Exercises
Now it is your turn. Evaluate the following single-variable expressions.
- \begin{align*}4x-9\end{align*} if \begin{align*}x\end{align*} is 20
- \begin{align*}5y+6\end{align*} if \begin{align*}y\end{align*} is 9
- \begin{align*}\frac{a}{4}-8\end{align*} if \begin{align*}a\end{align*} is 36
Take a few minutes to check your work with a friend.
II. Evaluate Multi-Variable Expressions with Given Values for the Variables
Algebraic expressions can have more than one variable. Look at the following examples of 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.
Example
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 expression.
\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 example where a given value is a fraction.
Example 4
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 by placing the number 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}\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 numbers.
\begin{align*}6 + 27\!\\ 33\end{align*}
Our answer is 33.
1F. Lesson Exercises
Evaluate the following expressions using the given values.
- \begin{align*}ab+7\end{align*} when \begin{align*}a\end{align*} is 9 and \begin{align*}b\end{align*} is 8
- \begin{align*}xy+zx\end{align*} when \begin{align*}x\end{align*} is 2, \begin{align*}y\end{align*} is 5 and \begin{align*}z\end{align*} is 7
- \begin{align*}xy+x\end{align*} when \begin{align*}x\end{align*} is \begin{align*}\frac{1}{4}\end{align*}, and \begin{align*}y\end{align*} is \begin{align*}\frac{4}{5}\end{align*}
Now check your answers with a partner. Be sure that your work is accurate. Did you remember how to add the fractions?
III. Use Given Expressions to Analyze and Solve Real-World Problems
When examining situations in the real world, we can use a variable to describe an unknown quantity. For example, let’s say that the bookstore sells boxes of pencils with 12 pencils in each box. Alicia bought several boxes of pencils. How many pencils did she buy all together?
In this situation, we know how many pencils are in each box, but we don’t know how many boxes Alicia bought. Whenever we don’t know a quantity, we can give it a variable. In this case, we can give the number of boxes she bought a variable, such as \begin{align*}n\end{align*}. Therefore, the expression \begin{align*}12n\end{align*} would represent the total number of pencils Alicia bought. When we find out the value of \begin{align*}n\end{align*} (the number of boxes), we will be able to evaluate the expression and find out the total number of pencils she bought. If we were to find out that \begin{align*}n = 6\end{align*}, we would know that she bought 6 boxes of pencils and we would evaluate our expression by substituting the value of our variable into our expression.
\begin{align*}12n &= 12(6)\\ 12n &= 72\end{align*}
Alicia bought 72 pencils.
Now let’s look at an example with two variables.
Example
The expression \begin{align*}8x + y\end{align*} can be used to describe how much money a waiter makes each day, if he earns $8.00 per hour, if \begin{align*}x =\end{align*} the number of hours worked and \begin{align*}y =\end{align*} the amount made in tips. On Monday, Mark worked 6 hours and made $12 in tips. How much did he make in all?
In this problem, there are two variables representing unknown quantities. \begin{align*}x\end{align*} is used to represent the number of hours a waiter works and \begin{align*}y\end{align*} is used to represent the amount of tips. We know the value of \begin{align*}x\end{align*} (6 hours), and we also know the value of \begin{align*}y\end{align*} ($12), so we can evaluate our expression to find out how much Mark made on Monday.
\begin{align*}8x + y &= 8(6) + 12 \\ 8x + y &= 48 + 12 \\ 8x + y &= 60 \end{align*}
The answer is $60.00
Remember to add the dollar sign, because the answer is in dollars.
The example that you read in the introduction is an example of an expression for a real-life situation.
IV. Write Variable Expressions to Represent and Solve Real-World Problems
Now that we know how to evaluate expressions, both alone and in relation to real-world problems, let’s use some familiar situations to write and evaluate our own variable expressions. The key to writing variable expressions is analyzing the information you have, identifying the information you need, and recognizing the terms that indicate which mathematical operation to use.
Let’s imagine that Ralph is a baker who makes \begin{align*}l\end{align*} number of loaves of bread each day.
- If he uses 5 cups of flour for each loaf, the total amount of flour he uses each day would be represented as \begin{align*}5l\end{align*}.
- If he only sells half the loaves he makes, then he sells \begin{align*}\frac{l}{2}\end{align*}.
- In 2 days, 3 days, 4 days, Ralph would make \begin{align*}2l, 3l\end{align*}, and \begin{align*}4l\end{align*} loaves of bread.
Now let’s go back to the introduction problem and apply all that we have learned about variable expressions to Kelly and her hiking.
Real Life Example Completed
The Pre-Trip Meeting
Here is the original problem once again. Reread this problem and underline all of the important information.
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 sat 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 skills that they would learn. The leaders said that the students would be hiking 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 distance traveled depends on the group,” said Scott one of the leaders. “If your group works well together, then 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*} mile 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
\begin{align*}x\end{align*} number of miles hiked per hour
Kelly wants to figure out the number of miles possible when the rate of hiking is in 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.
First, let’s write an expression to show the number of hours hiking times the possible number of miles in an hour. Since the number of miles per hour can vary, we use the variable \begin{align*}x\end{align*} for this rate.
\begin{align*}6x\end{align*}
The range of hiking rates can vary from \begin{align*}\frac{1}{2}\end{align*} mile per hour on the steepest terrain, to 2 miles per hour on flat terrain.
Kelly substitutes one-half for \begin{align*}x\end{align*} to figure out about how many miles they will cover if all six hours are spent 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. Next, Kelly looks at medium terrain. She substitutes 1 mile per hour into the expression for \begin{align*}x\end{align*}.
\begin{align*}6(1)=6\end{align*} miles on medium terrain
Finally, Kelly looks at flat terrain. She substitutes 2 miles per hour 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 hour, she figures that the group may cover a little less than 7 and one-half miles per day, or a little more, but it is probably a good middle estimate for distance covered per day.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- 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.
Technology Integration
Khan Academy Evaluating an Expression
James Sousa, Evaluating Algebraic Expressions
Time to 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*}
Directions: Evaluate each multi-variable expression.
16. Evaluate \begin{align*}xy + 9y\end{align*} if \begin{align*}x = 3\end{align*} and \begin{align*}y = 4\end{align*}
17. Evaluate \begin{align*}2x+3y\end{align*} if \begin{align*}x=10\end{align*} and \begin{align*}y=5\end{align*}
18. Evaluate \begin{align*}4y + 3y - 2z\end{align*} if \begin{align*}y = 2\end{align*} and \begin{align*}z = 4\end{align*}
19. 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
20. 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
Directions: Use what you learned about variables and real-life situations for the next problems. Choose a variable to represent the number and write an algebraic expression for the following phrases.
21. 19 decreased by a number
22. 4 less than the product of 4 and a number
23. 30 more than a number
24. 12 more than 3 times a number
25. A number divided by seven
26. A librarian has 4 times as many mystery books as romances. She lends out 12 mysteries. How many mysteries does she have now if she started with 15 romances?
27. In Saturday’s basketball game, Roman scored a fourth of his team's points. If the team scored 48 points total, how many points did Roman score? Write an expression and solve.
28. At the garden show daffodil bulbs cost $3 and tulip bulbs cost $4. Latoya buys 7 tulip bulbs and twice as many daffodil bulbs as tulips bulbs. How much does she spend total? Write an expression and solve.