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Single Variable Equations from Verbal Models

Write equations from key words and verbal phrases.

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Single Variable Equations from Verbal Models
License: CC BY-NC 3.0

Kelvin has twice as many chickens in his chicken coop as Murray has in his. If Kelvin has 60 chickens in his coop, write an equation to represent \begin{align*}c\end{align*}, the number of chickens in Murray’s chicken coop.

In this concept, you will learn to write single variable equations from verbal models.

Writing Single Variable Equations from Verbal Models

Changing a word problem into an equation can often help you in solving the problem. Carefully read the question to determine if the equation has addition, subtraction, multiplication, or division.

Here is an example.

Write “four times a number is twelve” as an equation.

First, let \begin{align*}x\end{align*} be “a number.”

Next, consider that the word “times” means you multiply.

\begin{align*}4 \text { times } x\end{align*}

\begin{align*}4x\end{align*}

Then, consider that the word “is” means equals.

\begin{align*}4x \text { is } 12\end{align*}

The equation is \begin{align*}4x = 12\end{align*}.

Here is another example.

Write “seven less than a number is fourteen” as an equation.

First, let \begin{align*}x\end{align*} be “a number.”

\begin{align*}7 \text{ less than } x \text { is } 14\end{align*}

Next, consider that “less than” means subtraction, but be careful about the order.

\begin{align*}x - 7 \text { is } 14\end{align*}

Then, consider that the word “is” means equals.

\begin{align*}x - 7 = 14\end{align*}

The equation is \begin{align*}x - 7 = 14\end{align*}.

Examples

Example 1

Earlier, you were given a problem about the chicken coops.

Kelvin has 60 chickens, which is twice as many chickens as Murray has. How can Kelvin write a single variable equation to represent his number of chickens?

First, simplify the language.

Kelvin has 60 chickens and this is 2 times \begin{align*}c\end{align*}, the number of chickens Murray has.

Next, consider that “is” means equals, and “times” means multiplication.

\begin{align*}\begin{array}{rcl} 60 &=& 2 \times c \\ 60 &=& 2c \end{array}\end{align*}

The equation \begin{align*}60 = 2c\end{align*} represents the number of chickens in Murray’s coop.

Example 2

Carrie made 3 liters of lemonade for a party. After the party, she had 0.5 liters of lemonade left. Write an equation to represent \begin{align*}n\end{align*}, the number of liters of lemonade that her guests drank.

Use a number, an operation sign, a variable, or an equal sign to represent each part of the problem. Since the question tells you how many liters of lemonade were left after the party, this will be a subtraction equation.

Carrie started with 3 liters, and \begin{align*}n\end{align*} is the number of liters that the guest drank. So, \begin{align*}3 - n\end{align*} is how much lemonade there was after the party, 0.5 liters.

For this problem, it may help to write an equation in simplified words and then translate those words into an algebraic equation.

\begin{align*}\begin{array}{rcl} \text{(number of liters made)} - \text{(number of liters guests drank)} &=& \text{(number of liters left)}\\ 3 - n &=& 0.5 \end{array}\end{align*}

The equation is \begin{align*}3 - n = 0.5\end{align*}.

Example 3

Write an equation for the following phrase: six and a number is twenty.

First, let \begin{align*}x\end{align*} be “a number.”

\begin{align*}6 \text { and } x \text { is } 20\end{align*}

Next, consider that “and” means addition.

\begin{align*}6 + x \text { is } 20\end{align*}

Then, consider that the word “is” means equals.

\begin{align*}6 + x = 20\end{align*}

The equation is \begin{align*}6 + x = 20\end{align*}.

Example 4

Write an equation for the following phrase: eighteen divided by a number is three.

First, let \begin{align*}x\end{align*} be “a number.”

\begin{align*}18 \text { divided by } x \text { is } 3\end{align*}

Next, you know this involves division.

\begin{align*}\frac{18}{x} \text { is } 3\end{align*}

Then, consider that the word “is” means equals.

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

The equation is \begin{align*}\frac{18}{x} = 3\end{align*}.

Example 5

Write an equation for the following phrase: five times a number is twenty-five.

First, let \begin{align*}y\end{align*} be “a number.”

\begin{align*}5 \times y \text { is } 25\end{align*}

Next, consider that the word “times” means you multiply.

\begin{align*}5y \text { is } 25\end{align*}

Then, consider that the word “is” means equals.

\begin{align*}5y = 25\end{align*}

The equation is \begin{align*}5y = 25\end{align*}.

Review

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.

Write an algebraic expression for each situation below.

  1. 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.
  2. 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.
  3. 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.
  4. 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.

Write an algebraic equation for each word problem below.

  1. 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.
  2. 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.

Review (Answers)

To see the Review answers, open this PDF file and look for section 7.3.

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Vocabulary

TermDefinition
Algebraic Expression An expression that has numbers, operations and variables, but no equals sign.
Equation An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.
Expression An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.
Variable A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.

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