7.3: Single Variable Equations from Verbal Models
Have you ever known anyone with a stamp collection? Take a look at this dilemma.
Kelvin has twice as many stamps in his collection as Murray has in his. If Kelvin has 60 stamps in his collection, write an equation to represent \begin{align*}m\end{align*}, the number of stamps in Murray's collection.
To do this, you will need to know how to write single variable equations from verbal models. Pay attention to this Concept and you will learn all that you need to know.
Guidance
What is the difference between an equation and an expression? Well, an expression is a phrase without an equal sign. This means that the variable in an expression can be changed and the expression can be evaluated differently. An equation has an equal sign. Therefore one side of an equation is equal to a value on the other side.
Now, let's examine how to write equations.
The same key words that helped you write expressions may also help you write equations. Here are some additional key words that you may find helpful.
Key Words for Addition or Multiplication Equations | Key Words for Subtraction Equations | Key Words for Division Equations |
---|---|---|
how many all together how many in all how many total |
how many more how many fewer how many left how much change |
how many in each |
Take a few minutes to write down these key words and phrases in your notebook.
Now let's apply this information.
4 times a number is twelve.
First, notice that we have the word “times” which means to multiply. We also have the word “is” which is a key word for equals.
\begin{align*}4 \ times \ x=12\end{align*}
The equation is \begin{align*}4x=12\end{align*}.
Seven less than a number is fourteen.
First, notice we have the words “less than” which means subtraction. Then we have the word “is” which is a key word for equals.
\begin{align*}x-7=14\end{align*}
The answer is \begin{align*}x-7=14\end{align*}.
Write an equation for each verbal phrase.
Example A
Six and a number is twenty.
Solution: \begin{align*}6 + x = 20\end{align*}
Example B
Eighteen divided by a number is three.
Solution: \begin{align*}18 \div x = 3\end{align*}
Example C
Five times a number is twenty-five.
Solution: \begin{align*}5y = 25\end{align*}
Here is the original problem once again.
Kelvin has twice as many stamps in his collection as Murray has in his. If Kelvin has 60 stamps in his collection, write an equation to represent \begin{align*}m\end{align*}, the number of stamps in Murray's collection.
Use a number, an operation sign, a variable, or an equal sign to represent each part of that problem. Since Kelvin has 60 stamps in his collection, represent the number of stamps in Kelvin's collection as 60.
\begin{align*}& \underline{Kelvin} \ \underline{has} \ \underline{twice \ as \ many \ stamps ... as \ Murray}.\\ & \downarrow \qquad \quad \downarrow \qquad \qquad \qquad \downarrow\\ & 60 \qquad \ = \qquad \qquad \quad \ 2m\end{align*}
This equation \begin{align*}60=2m\end{align*}, represents \begin{align*}m\end{align*}, the number of stamps in Murray's collection.
Vocabulary
Here are the vocabulary words in this Concept.
- Expression
- a number sentence with variables, numbers and operations.
- Variable
- a letter used to represent an unknown quantity.
- Algebraic Expression
- a combination of multiple variables, numbers and operations.
- Equation
- a number sentence with an equal sign where the quantity on one side of the equals is the same as the quantity on the other side.
Guided Practice
Here is one for you to try on your own.
Carrie made 3 liters of lemonade for a party. After the party, she had 0.5 liter of lemonade left. Write an equation to represent \begin{align*}n\end{align*}, the number of liters of lemonade that her guests drank.
Answer
Use a number, an operation sign, a variable, or an equal sign to represent each part of that problem. Since the question tells us how many liters of lemonade were left after the party, write a subtraction equation.
Since she had 0.5 liter of lemonade left, \begin{align*}n\end{align*} is the number of liters that were drunk at the party. For this problem, it may help to write an equation in words and then translate those words into an algebraic equation.
\begin{align*}& (\text{number of liters made}) \ - \ (\text{number of liters guests drank}) \ = \ (\text{number of liters left})\\ & \qquad \downarrow \qquad \qquad \qquad \qquad \ \downarrow \qquad \qquad \qquad \downarrow \qquad \qquad \qquad \qquad \ \downarrow \qquad \qquad \quad \downarrow\\ & \qquad \ 3 \qquad \qquad \qquad \qquad - \qquad \qquad \ \ \ \ \ n \qquad \qquad \qquad \quad \ \ = \qquad \qquad \ 0.5\end{align*}
This equation, \begin{align*}3-n=0.5\end{align*}, represents \begin{align*}n\end{align*}, the number of liters of lemonade that Carrie's guests drank during the party.
Video Review
Here is a video for review.
Practice
Directions: Write an equation for each verbal model.
1. Ten times a number is thirty.
2. Five times a number is fifteen.
3. A number and seven is eleven.
4. A number divided by three is twelve.
5. A number and eighteen is thirty.
6. A number divided by twelve is fourteen.
7. Seven times a number is forty - nine.
8. A number divided by thirteen is seven.
9. Eight times a number is equal to sixty - four.
Directions: Write an algebraic expression for each situation below.
10. Arturo has 8 fewer stickers in his collection than Julissa has in hers. Let \begin{align*}j\end{align*} represent the number of stickers in Julissa's collection. Write an expression to represent the number of stickers in Arturo's collection.
11. Let \begin{align*}c\end{align*} represent the number of cookies on a plate. Three friends share all the cookies on the plate equally. Write an expression to represent the number of cookies each friend has after they are shared equally.
12. Carly is twice as old as her sister. Let \begin{align*}s\end{align*} represent her sister's age in years. Write an expression to represent Carly's age in years.
13. The length of a rectangle is 3 inches longer than its width. Let \begin{align*}w\end{align*} stand for the width in inches. Write an expression to represent the length in inches.
Directions: Write an algebraic equation for each word problem below.
14. The chorus teacher divides all the students in the chorus into 3 equal groups. Each of the groups has 6 students in it. Write an equation that could be used to represent \begin{align*}n\end{align*}, the total number of students in the chorus.
15. Matt's dog weighs 30 pounds. His dog weighs 20 pounds more than his cat. Write an equation that could be used to represent \begin{align*}c\end{align*}, the weight, in pounds, of Matt's cat.
Image Attributions
Here you'll learn to write single variable equations from verbal models.