2.1.2: Sentences as Single Variable Equations

Sentences as Single-Variable Expressions

Figure \(\PageIndex{1}\)

Mrs. Hughes knows that students dread final exams, so this semester she is going to allow her cooking class to bake cakes for its final exam. She plans to divide the 24 members of the class into six groups of four, and each group will bake a cake as their final grade. Mrs. Hughes is making a list of all the items she needs to purchase for the cakes. Eggs are at the top of her list because each group will need six eggs. There are 12 eggs in a dozen. How can Mrs. Hughes use this information to determine how many dozens of eggs the class will need?

In this concept, you will learn how to write sentences as single-variable equations.

Writing Sentences as Single-Variable Equations

Expressions contain some combination of numbers, variables and operations, but does not have an equals sign. When you have an equals sign, you have an equation not an expression.

An equation has an equal sign. One side of the equation equals the other side of the equation.

The following is an equation.

5+9=14

Here five plus nine is equal to fourteen. The quantity on one side of the equal sign is the same as the quantity on the other side of the equal sign.

You can also have equations with variables in them. When you have a variable in an equation, there is an unknown quantity.

Take a look at the following equation which contains a variable.

Five plus an unknown number is equal to fifteen.

To write an equation for this phrase, start by working through the problem from the left to the right.

First, write the expression for the initial part of the equation, “Five plus.”

5+

Next, let the “unknown number” be the variable x.

x

Then, express “Is equal to fifteen” with an equal sign and number.

=15

The answer is 5+x=15

Here is another equation.

Six less than a number is equal to ten.

First, write the expression for “Six less than.” Since this is “six less than” the order is reversed. It becomes some number or quantity minus 6.

−6

Next, let the “unknown number” be the variable x.

x

Then, express “Is equal to ten” with an equal sign and number.

=10

The answer is x−6=10.

Here is an example that uses multiplication.

The product of three and a number is thirty.

First, note that product means multiplication.

x

Next, identify the numbers involved on the left side of the equal sign.

3

Then, let the “unknown number” be the variable y.

y

Finally, express “Is 30” with an equal sign and number.

=30

The answer is 3y=30.

Examples

Example 6.2.1

Earlier, you were given a problem about Mrs. Hughes and her delicious final exams.

Before Mrs. Hughes goes to the grocery store, she has to determine how many dozens of eggs to purchase for her cooking class, which will be baking cakes as final exams. The class has been divided into six groups of four students. Mrs. Hughes is writing a list of items she needs to purchase for the cakes. One of the items is eggs. Each group needs six eggs for its cake, which is 36 eggs together. How can Mrs. Hughes use a single variable equation to decide how many dozens of eggs she needs to purchase?

Solution

First, identify the key information.

• The groups need a combined total of 36 eggs.
• There are 12 eggs in a dozen.

Next, express the situation in an equation.

Thirty-six eggs divided by 12 is an unknown number.

Then, write the first part of the equation.

36÷12

Finally, let y represent the unknown dozens of eggs she will need.

y

Now, write the equation.

36÷12=y

The answer is 3.

Mrs. Hughes needs to purchase three dozen eggs for the class to bake cakes.

Example 6.2.2

Write a single-variable equation for “six times a number divided by two is equal to four.”

Solution

Work through the problem from the left to the right.

First, note that “Six times” is multiplication.

6x

Next, let y represent the unknown number.

y

Then, express “is divided by 2.”

(6y)/2

Finally, express “is four” with an equal sign and number.

=4

The answer is (6y)/2=4.

Example 6.2.3

Write a single-variable equation for "fifteen divided by an unknown number is three."

Solution

To write an equation for this phrase, start by working your way through the problem from the left to the right.

First, identify the first part of the phrase – “Fifteen divided by”

15/(?)

Next, let y represent the unknown number.

y

Then, express “is three” with an equal sign and number.

=3

The answer is 15/y=3.

Example 6.2.4

Write a single-variable equation for "six times an unknown number is thirty-six."

Solution

First, note that “Six times” is multiplication.

6x

Next, let y represent the unknown number.

y

Then, express “is thirty-six” with an equal sign and number.

=36

The answer is 6y=36.

Example 6.2.5

Write a single-variable equation for "fifteen and twelve is an unknown number."

Solution

First, identify the numbers involved.

15 and 12

Next, identify the operation involved.

“and” means addition

+

Then, let y represent the unknown number.

y

Finally, express “is an unknown number” with equal signs and the variable y.

=y

The answer is 15+12=y.

Review

Write each phrase as a single-variable equation.

1. Five less than a number is fifteen.
2. The sum of a number and six is eighteen.
3. Twenty divided by a number is four.
4. Sixteen less than a number is four
5. Twelve and a number is twenty.
6. The product of six and a number is forty-two.
7. Eight times a number is forty.
8. Ten less than a number is twenty-one.
9. A number divided by two is seven.
10. A number times four is forty-eight.
11. An unknown divided by two is fourteen.
12. Twelve times an unknown number is sixty.
13. Fourteen divided by an unknown number is seven.
14. Five and a number is equal to fifty - three.
15. Ten less than a number is seventeen.

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.3.

Additional Resources

Video:

Practice: Sentences as Single Variable Equations

Real World Application: Where the Cassowary Roams