1.10: Write and Evaluate Variable Expressions for Given Situations
“Here is a change that is great,” Cameron the eighth grade President said to the Student Council on Wednesday afternoon. “We can fundraise during lunch by selling cookies and we don’t need permission.”
“Really, that is a change,” Marcy commented.
The students at the middle school had always been able to fundraise, but they needed to get special permission. This was especially true regarding food and regarding lunch time. This news was greeted by some excitement and a lot of talking at the meeting.
“Hold on everybody,” Cameron shouted over the crowd. “Let’s talk about this.”
The group quieted down and Cameron began.
“We need $350.00 for the first autumn dance. We have $50.00 in our account right now. I figure that if we can see cookies for 25 cents each, then we can make about $30.00 per week in the two lunches,” Cameron proposed.
“I think we’ll make double that,” Tracy said smiling.
“Well maybe, but if we make the $30.00 then we will have the money for the autumn dance in no time. Just do the math!”
Did you do the math? This Concept is about writing expressions. How long will it take based on Cameron’s plan? How long if Tracy is correct?
Guidance
Did you know that you can write a variable expression from a verbal expression and then evaluate that variable expression by using a given value?
You will have to work as a detective to figure out what different words mean. Once you know what those words mean, you will be able to write different variable expressions.
Let’s start by looking at some mathematical operations written as words.
Addition
Sum
Plus
Increased by
More
Subtraction
Difference
Less than
Take away
Multiplication
Product
Times
Division
Quotient
Split up
This list does not include ALL of the ways to write the operations, but it will give you a good place to start.
Take a few minutes and write these words down in your notebook.
Now we can look at the following chart which starts with a verbal phrase and writes it as a variable expression.
Verbal Phrase | Variable Expression |
---|---|
Three minus a number |
\begin{align*}3 - x\end{align*} |
A number increased by seven | \begin{align*}n + 7\end{align*} |
The difference between an unknown quantity and twenty-six | \begin{align*}s - 26\end{align*} |
A number decreased by nine | \begin{align*}w - 9\end{align*} |
Ten times a number plus four | \begin{align*}10f + 4\end{align*} |
Notice that words like “a number” and “an unknown quantity” let us know that we need to use a variable.
Look at this problem.
Evaluate four times a number minus five, when the number is four.
First, we need to write an expression. We know that there are two operations in this expression. The word “times” tells us that we have multiplication and the word “minus” tells us that we have subtraction. We also know that the value of the variable is four.
\begin{align*}4n-5, \ when \ n \ is \ 4\end{align*}
Next, we can substitute the given value in for \begin{align*}n\end{align*} and evaluate the expression.
\begin{align*}& 4(4)-5\\ & 16-5\\ & 11\end{align*}
The answer is 11.
We can write variable expressions for real-world situations too.
Grace is saving money to purchase a new bike that costs one hundred seventy-five dollars. Grace has twenty-five dollars and is also saving twenty dollars each week. Write and solve a variable expression to determine the number of weeks “\begin{align*}w\end{align*}” it will take Grace to save for her new bike.
First, add the amount that Grace has already saved, $25.00, to the amount that she plans on saving each week. Because the number of weeks is unknown, we multiply the amount she is planning on saving each week, $20.00 by the variable “\begin{align*}w\end{align*}”. The “\begin{align*}w\end{align*}” represents the number of weeks. Set the expression equal to Grace’s goal of $175.00.
\begin{align*}25.00 + 20.00w = 175.00\end{align*}
Now we know that she needs to earn an amount of money that equals $175.00. Because she has already earned 25.00, we can subtract that from the total she needs.
\begin{align*}175.00 - 25.00 = \$150.00\end{align*}
We are left with the expression.
\begin{align*}20.00w = 150.00\end{align*}
Solve for “\begin{align*}w\end{align*}” by completing the inverse of multiplication. Since the number of weeks “\begin{align*}w\end{align*}” times 20.00 equals 150.00, divide 150.00 by \begin{align*}20.00w\end{align*}.
\begin{align*}\frac{20.00w}{20.00}=\frac{150.00}{20.00}=7.5\end{align*}
Therefore, it will take Grace seven and one half weeks to save for her bike.
Evaluate each expression.
Example A
\begin{align*}6x + 5, \ when \ x \ is \ 12\end{align*}
Solution: \begin{align*}77\end{align*}
Example B
\begin{align*}-4x-5x, \ when \ x \ is \ 3\end{align*}
Solution: \begin{align*}-27\end{align*}
Example C
\begin{align*}-6y+8y, \ when \ y \ is \ 5\end{align*}
Solution: \begin{align*}10\end{align*}
Now let's go back to the dilemma from the beginning of the Concept.
To work on this problem, let’s first think about the given information and write an expression. We use Cameron’s information to do this.
There is $50.00 in the student council account.
Cameron thinks that they can save $30.00 per week.
The number of weeks needed to save the money is unknown that is our variable, \begin{align*}w\end{align*}.
Here is the expression.
\begin{align*}50 + 30w\end{align*}
Next, we can write an equation using this expression to solve for the number of weeks needed to make $350.00.
\begin{align*}50 + 30w = 350\end{align*}
We can solve it by using mental math.
\begin{align*}w = 10\end{align*}
It will take 10 weeks based on Cameron’s proposal.
If Tracy is correct and the students are able to make twice as much money, then it will take them 5 weeks to make the money.
Vocabulary
- Variable Expression
- a group of numbers, operations and variables without an equal sign.
- Variable
- a letter used to represent an unknown number
- Constant
- a number in an expression that does not have a variable.
- Verbal Expression
- using language to write a mathematical expression instead of numbers, symbols and variables.
Guided Practice
Here is one for you to try on your own.
Evaluate a number squared plus six when the number is eight.
Solution
First, notice that we have the word “squared” that lets us know that we will be working with a power. Then we have the word “plus” so we know that our second operation is addition. The unknown value is eight.
\begin{align*}x^2+6 \ when \ x \ is \ 8\end{align*}
Next, we substitute the 8 in for the value of the variable.
\begin{align*}& 8^2+6\\ & 64 + 6\\ & 70\end{align*}
The answer is 70.
Video Review
Practice
Directions: Evaluate each verbal phrase.
- The sum of a number 12 and twelve.
- The difference between a number 12 and eight.
- Three times a number 12
- A number 12 squared plus five
- A number 12 divided by two plus seven
- Four times the quantity of a number 12 plus six
- A number 12 times two divided by four
- A number 12 times six plus the same number times two
- A number 12 squared plus seven take a way four
- A number 12 divided by three plus twelve
Directions: Evaluate each expression if \begin{align*}x=2,y=-4\end{align*}
11. \begin{align*}4x+6y-4\end{align*}
12. \begin{align*}-6y+x-9\end{align*}
13. \begin{align*}-4xy-6xy\end{align*}
14. \begin{align*}9xy+8\end{align*}
15. \begin{align*}-8x-12y+15\end{align*}
Algebraic Expression
An expression that has numbers, operations and variables, but no equals sign.Verbal Expression
A verbal expression (or verbal model) uses words to decipher the mathematical information in a problem. An equation can often be written from a verbal model.Image Attributions
Here you'll learn to write and evaluate variable expressions for given situations.
Concept Nodes:
Algebraic Expression
An expression that has numbers, operations and variables, but no equals sign.Verbal Expression
A verbal expression (or verbal model) uses words to decipher the mathematical information in a problem. An equation can often be written from a verbal model.