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Single Variable Expressions

Use symbols and operations to understand and define variables.

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Write and Evaluate Variable Expressions for Given Situations

License: CC BY-NC 3.0

Josephine, a local artist, has been commissioned to enlarge a quilt hanging. The new quilt cannot exceed 7.5 feet in length or 6.5 feet in width due to the space available for hanging. After much thought, Josephine has decided to increase the size of the 4 foot by 3 foot quilt by adding a uniform, quilted strip around the existing one. Josephine thinks that a 1.25 feet wide strip will work, but how can Josephine use a variable expression to figure this out?

In this concept, you will learn to write and evaluate variable expressions for given situations.

Variable Expressions for Given Situations

Words can be translated into mathematical expressions to represent real-world problems. Then, mathematical operations can be used to solve the problem. You must be able to recognize the operations being presented by the words and be able to use symbols to display them. All unknown quantities can be represented by variables.

Let’s look at an example.

Complete the following table with variable expressions having either one or two variables as needed.

Words to be Translated Translations Variable Expression
“Six times a number decreased by 7”
“The sum of Sam’s age and Joe’s age”
“The value, in cents, of \begin{align*}y\end{align*} dimes”
“Three times the length less twice the width”
Words to be Translated Translations Variable Expression
“Six times a number decreased by 7” \begin{align*}\underbrace{\text{Six times a number}}_{{\color{brown}6n}} \ \ \underbrace{\text{decreased by}}_{{\color{brown}-}} \ \ \underbrace{\text{seven}}_{{\color{brown}7}}\end{align*} \begin{align*}6n-7\end{align*}
“The sum of Sam’s age and Joe’s age” \begin{align*}\underbrace{\text{The sum of}}_{{\color{blue}+}} \ \ \underbrace{\text{Sam's age}}_{{\color{blue}s}} \ \ \text{and} \ \ \underbrace{\text{Joe's age}}_{{\color{blue}j}}\end{align*}  \begin{align*}s+j\end{align*}
“The value, in cents, of \begin{align*}y\end{align*} dimes” \begin{align*}\underbrace{\text{The value in cents}}_{{\color{green}0.10}} \ \ \text{of} \ \ \underbrace{y\text{ dimes}}_{{\color{green}y}}\end{align*} \begin{align*}0.10y\end{align*}
“Three times the length less twice the width” \begin{align*}\underbrace{\text{Three times the length}}_{{\color{red}3l}} \ \ \underbrace{\text{less}}_{-} \ \ \underbrace{\text{twice the width}}_{{\color{red}2w}}\end{align*} \begin{align*}3l-2w\end{align*}

Once you have written a variable expression to represent the words, you can evaluate the expression for a given value of the variable or values of the variables.

Let’s look at this problem.

Evaluate “Four times a number reduced by “five” when the number is 4.

First, write a variable expression.

There are two operations presented by the words. The word “times” tells you there is multiplication and the words “reduced by” tells you there is subtraction.

\begin{align*}\underbrace{\text{Four times a number}}_{4n} \quad \underbrace{\text{reduced by}}_{-} \quad \underbrace{\text{five}}_{5} \end{align*}

The variable expression is \begin{align*}4n-5\end{align*}.

First, evaluate the expression when \begin{align*}n=4\end{align*}.

Next, substitute the value \begin{align*}n=4\end{align*} into the expression.

\begin{align*}4(4)-5\end{align*}

Next, multiply \begin{align*}4(4)=16\end{align*} to clear the parenthesis. Write the new expression.

\begin{align*}16-5\end{align*}

Then, subtract:

\begin{align*}16-5=11\end{align*}

The answer is 11.

Examples

Example 1

Earlier, you were given a problem about Josephine and her quilt enlargement.

In order to enlarge the quilt she has to add two strips of the same width to the width of the quilt and two more strips of the same width to the length of the quilt. To figure out if a strip 1.25 feet wide will work, Josephine has to write two variable expressions – one for the length and one for the width.

For the length of 4 feet:

First, name the variable. Let “\begin{align*}x\end{align*}” represent the width of the strip.

Next, include any operation on the variable. \begin{align*}2x\end{align*}

Next, include any constant associated with the variable: 4

Next, represent the operation between the constant and the variable by its symbol. +

Then, write the variable expression for the length.

The answer is \begin{align*}4+2x\end{align*}.

For the width of 3 feet:

First, name the variable. Let “\begin{align*}x\end{align*}” represent the width of the strip.

Next, include any operation on the variable. \begin{align*}2x\end{align*}

Next, include any constant associated with the variable: 3

Next, represent the operation between the constant and the variable by its symbol. +

Then, write the variable expression for the length.

The answer is \begin{align*}3+2x\end{align*}.

The space for hanging the quilt is 7.5 feet in length and 6.5 feet in width.

First, substitute \begin{align*}x=1.25\end{align*} into the variable expression for the length.

\begin{align*}4+2(1.25)\end{align*}

Next, multiply \begin{align*}2(1.25)=2.50\end{align*} to clear parenthesis.

\begin{align*}4+2.50\end{align*}

Then, add:

\begin{align*}4+2.50=6.50\end{align*}

The answer is 6.5.

First, substitute \begin{align*}x=1.25\end{align*} into the variable expression for the width.

\begin{align*}3+2(1.25)\end{align*}

Next, multiply \begin{align*}2(1.25)=2.50\end{align*} to clear parenthesis.

\begin{align*}3+2.50\end{align*}

Then, add:

\begin{align*}3+2.50=5.50\end{align*}

The answer is 5.5.

Josephine can hang her enlarged quilt in the available space.

Example 2

Grace is saving money to purchase a new bike that costs one hundred eighty-five dollars. She has twenty-five dollars saved already and plans to save twenty dollars of her babysitting money each week. Write a variable expression to model the problem.

First, name the variable. Let ‘\begin{align*}w\end{align*}’ represent the number of weeks she has to save twenty dollars.

Next, include the dollar value with the variable.

\begin{align*}20w\end{align*}

Next, add the money she has already saved. +25

\begin{align*}20w + 25\end{align*}

Then, evaluate the variable expression

\begin{align*}20w + 25\end{align*} for \begin{align*}w=7\end{align*} and \begin{align*}w=8\end{align*}.

If one of these given values result in the variable expression being $185.00, then you know that Grace has to save twenty dollars for that number of weeks to have enough money to buy her new bike.

First, substitute \begin{align*}w=7\end{align*} into the expression.

\begin{align*}20(7)+25\end{align*}

Next, multiply: \begin{align*}20(7)=140\end{align*} to clear the parenthesis.

\begin{align*}140+25\end{align*}

Then, add:

\begin{align*}140+25=165\end{align*}

The answer is $165.00

First, substitute \begin{align*}w=8\end{align*} into the expression.

\begin{align*}20(8)=25\end{align*}

Next, multiply: \begin{align*}20(8)=160\end{align*} to clear the parenthesis.

\begin{align*}160+25\end{align*}

Then, add:

\begin{align*}160+25=185\end{align*}

The answer is $185.00.

Grace must save $20.00 for eight weeks to have enough money to buy the bike.

Example 3

Write a variable expression to model “Twice the length increased by three times the width” and evaluate it when the length is 12 and the width is 9.

First, name the variables. Let “\begin{align*}l\end{align*}” represent the length. Let “\begin{align*}w\end{align*}” represent the width.

Next, include any operations on the variables.

\begin{align*}2l\end{align*} and \begin{align*}3w\end{align*}

Next, represent the operation between the variables by its symbol. +

Then, write the variable expression.

\begin{align*}\underbrace{\text{Twice the length}}_{2l} \quad \underbrace{\text{increased by}}_{+} \quad \underbrace{\text{three times the width}}_{3w}\end{align*} 

The answer is \begin{align*}2l + 3w\end{align*}.

First, substitute the values \begin{align*}l=12\end{align*} and \begin{align*}w=9\end{align*} into the expression.

\begin{align*}2(12)+3(9)\end{align*}

Next, multiply \begin{align*}2(12)=24\end{align*} to clear the parenthesis.

\begin{align*}24+3(9)\end{align*}

Next, multiply \begin{align*}3(9)=27\end{align*} to clear the parenthesis.

\begin{align*}24+27\end{align*}

Then, add:

\begin{align*}24+27=51\end{align*}

The answer is 51.

Example 4

Jake and his Uncle Mark were sitting on the front step discussing what Jake was learning in Math this year. Jake looked up at his uncle and said, “Uncle Mark, do you know what one-fourth my age will be eight years from now?”

First, name the variable. Let “\begin{align*}a\end{align*}” represent Jake’s age now.

Next, write the expression to represent “\begin{align*}a\end{align*}” eight years from now.

\begin{align*}a+8\end{align*}

Then, include any operation performed on the variable. \begin{align*}\frac{1}{4}( \ )\end{align*}

\begin{align*}\frac{1}{4}(a+8)\end{align*}

Evaluate the expression when \begin{align*}a=16\end{align*}.

First, substitute the value \begin{align*}a=16\end{align*} into the expression.

\begin{align*}\frac{1}{4}(16+8)\end{align*}

Next, perform the operation in the parenthesis. \begin{align*}16+8=24\end{align*}

\begin{align*}\frac{1}{4}(24)\end{align*}

Then, multiply \begin{align*}\frac{1}{4}(24)=6\end{align*} to clear the parenthesis.

The answer is 6.

Review

Evaluate each verbal phrase.

1. The sum of a number 12 and twelve.

2. The difference between a number 12 and eight.

3. Three times a number 12.

4. A number 12 squared plus five.

5. A number 12 divided by two plus seven.

6. Four times the quantity of a number 12 plus six.

7. A number 12 times two divided by four.

8. A number 12 times six plus the same number times two.

9. A number 12 squared plus seven take away four.

10. A number 12 divided by three plus twelve.

Evaluate each expression when \begin{align*}x=2\end{align*} and \begin{align*}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*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.10. 

Resources

 

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Vocabulary

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.

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  1. [1]^ License: CC BY-NC 3.0

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