# 2.1.3: Addition and Subtraction Phrases as Equations

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## Addition and Subtraction Phrases as Equations Figure $$\PageIndex{1}$$

Sal leads an informal cycling team of six people. He wants to register them for a race, but doesn’t know if there is enough space. The maximum number of allowed racers on the course is 138 cyclists. What is the maximum number of cyclists that can already be registered if the whole team can join the race? How can Sal write a single variable equation to represent this problem?

In this concept, you will learn how to write addition and subtraction phrases as single variable equations.

### Writing Addition and Subtraction Phrases as Equations

An expression connects numbers and/or variables with operations, such as addition, subtraction, multiplication, and division. Notice that an expression does not have an equal sign. The value of the variable in each expression can change, and you can evaluate an expression using any value for the variable.

• 50−2
• 4−a
• 12z
• 4x/3

In the expressions above, a,z, and x are variables.

An expression that includes one or more variables is called an algebraic expression. You can use algebraic expressions to represent words or phrases. To help you solve word problems, you need to translate words or phrases into operations, variables, or expressions. Let’s start with addition and subtraction phrases. Take a look at this chart.

 Addition Phrases Subtraction Phrases 1 plus a 1+a 4 less d 4−d 2 and b 2+b 6 less than g g−6 4 more than c c+3 h fewer than 7 7−h

The bolded words in the phrases above tell you if you should use addition or subtraction, and the order of the terms. Read the word problem carefully to figure out which operation makes the most sense.

Let’s look at an example.

Abdul has $5 more than Xavier has. Write an algebraic expression to show the number of dollars Abdul has. The phrase is “$5 more than Xavier.” Use a number, an operation sign, or a variable to represent each part of that phrase.

Let x stand for the number of dollars Xavier has. “More than” means you are adding.

x+5

So, the expression x+5 represents the number of dollars Abdul has.

Let’s look at another example.

Change “6 less than a number” into an algebraic expression.

Let “a number” be the variable x. “Less than” means you are subtracting. Be careful about the order, 6 will follow the variable.

x−6

Here is one more example.

Lian is x inches shorter than Hannah. Hannah is 65 inches tall. Write an algebraic expression to show Lian’s height in inches.

The phrase is “x inches shorter than Hannah.” You also know that Hannah’s height is 65 inches. “Shorter than” means you are subtracting. Be careful about the order, x will follow 65.

65−x

In a word problem, the word “is” means equals. When you see the word “is” you can set the expression equal to something. This is an equation.

### Examples

Example 1.3.1

Earlier, you were given a problem about Sal’s biking team.

Sal needs to register his team of six people, but the maximum number of allowed racers on the course is 138 cyclists. Sal needs to write a single variable equation to figure out the maximum number of cyclists that can already be registered so that his team of six can join.

Solution

First, let x be the maximum number of cyclists that can be registered. Six more than that will give you the maximum of 138 cyclists. When you can use the phrase “more than” you can use addition.

Example 1.3.2

Write an equation to represent the following phrase.

Four less than an unknown number is eighteen.

Solution

To figure this out, first let “an unknown number” be the variable x. Next, you know the operation is subtraction because of the key phrase “less than”.

So you can write x−4 since the four is being taken away from the unknown number.

The word “is” means equals, and “eighteen” is 18.

x−4=18

Example 1.3.3

Write an equation for the following phrase: a number plus five is ten.

Solution

Let x be “a number”, “plus” means addition, and “is” means equals.

Example 1.3.4

Write an equation for the following phrase: six more than a number is eighteen.

Solution

Let x be “a number”, “more than” means addition, and “is” means equals.

Example 1.3.5

Write an equation for the following phrase: fifteen less than a number is twenty.

Solution

Let x be “a number”, “less than” means subtraction (but be careful about order), and “is” means equals.

### Review

Write an expression for each phrase.

1. 5 more than a number
2. A number plus six
3. 8 and a number
4. Seven less than a number
5. Eight take a way four
6. Nine more than a number

Write a simple equation for each phrase.

1. Five less than a number is ten.
2. Eight take away four is a number.
3. Five and a number is twelve.
4. Sixteen less than an unknown number is eighty.
5. Twenty and a number is fifty - five.
6. A number and fifteen is forty.
7. A number and twelve is sixty.
8. Fifteen less than a number is ninety.
9. Sixty less than a number is eighty.

### Vocabulary

Term Definition
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.