7.1: Addition and Subtraction Phrases as Equations
Have you ever been to Boston? Have you ever spent the summer with your grandparents?
Kara and her twin brother Marc are going to be spending one month in Boston with their grandparents. They are very excited! Not only is it summer in Boston and the weather will be terrific, but Boston has a fantastic subway system called the T and they will be able to ride it to get all around.
Kara is very excited about visiting the museums, but Marc who is a huge baseball fan is hoping for a trip to Fenway Park before the month is up. Seeing the Red Sox play would be a huge bonus!
Kara has earned $125.00 babysitting and has some birthday money too.
Before the trip, she counted her savings and realized that she has $54.25.
You can write an equation to show the unknown in this dilemma.
This Concept will teach you how to write addition and subtraction phrases as single variable equations.
Guidance
An expression shows how numbers and/or variables are connected by operations, such as addition, subtraction, multiplication, and division. Notice that an expression does not have an equal sign. This is because the value of the variable in each expression can change, or you could say that we can evaluate an expression using different given values for the variable.
\begin{align*}& 50-2 && 4-a && 12z && \frac{4x}{3}\end{align*}
Three of the expressions above include variables, such as \begin{align*}a, z\end{align*}, and \begin{align*}x\end{align*}.
An expression that includes one or more variables is called an algebraic expression. Each variable in an algebraic expression can have a different value. Once again, you will not see an equal sign with an algebraic expression. We can use algebraic expressions to represent words or phrases.
Often in word problems or in other situations in math, you will be given a set of words or a phrase that you will need to rewrite as an expression. When you do this, you will be looking for words that mean different operations or things in math. This way you can write an expression that correctly represents the words or phrase.
We are going to start with addition and subtraction phrases. Take a look at this chart.
Addition Expressions | Subtraction Expressions | ||
---|---|---|---|
1 plus \begin{align*}a\end{align*} | \begin{align*}1+a\end{align*} | 4 less \begin{align*}d\end{align*} | \begin{align*}4-d\end{align*} |
2 and \begin{align*}b\end{align*} | \begin{align*}2+b\end{align*} | 6 less than \begin{align*}g\end{align*} | \begin{align*}g-6\end{align*} |
3 more than \begin{align*}c\end{align*} | \begin{align*}3+c\end{align*} | \begin{align*}h\end{align*} fewer than 7 | \begin{align*}7-h\end{align*} |
The bolded key words in the phrases above provide clues about whether or not you should write an addition or a subtraction expression. While key words can be a helpful guide, it is important not to rely on them totally. It is always most important to think about which operation makes the most sense for a particular situation.
Write these key words in your notebook and then continue with the Concept.
Abdul has $5 more than Xavier has. Let \begin{align*}x\end{align*} stand for the number of dollars 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.
\begin{align*}& \$ \underline{5} \ \underline{more \ than} \ \underline{Xavier}\\ & \downarrow \qquad \quad \downarrow \qquad \quad \ \downarrow\\ & \ 5 \qquad \ \ + \qquad \quad x\end{align*}
Notice that in this phrase, the key words “more than” means you should write an addition expression.
So, the expression \begin{align*}5+x\end{align*} represents the number of dollars Abdul has. We also could have written this as. \begin{align*}x+5\end{align*} because addition is commutative. That means, the order in which numbers are added does not matter.
Our answer is \begin{align*}x+5\end{align*}.
Here is another one.
6 less than a number
Notice the key words “less than” which means subtraction.
A number means a variable.
Because it says “6 less than a number,” the 6 will follow the variable.
\begin{align*}x-6\end{align*}
Our answer is \begin{align*}x-6\end{align*}.
Remember, sometimes writing an expression is not as simple as relying on key words.
Lian is \begin{align*}x\end{align*} inches shorter than Hannah. Hannah is 65 inches tall. Write an algebraic expression to show Lian's height in inches.
The phrase is “\begin{align*}x\end{align*} inches shorter than Hannah.” You also know that Hannah's height is 65 inches.
There are no key words, so you need to think about what makes sense. If Lian is shorter than Hannah, her height will be less than 65 inches. So, write a subtraction expression. Use 65 to represent Hannah's height. Since Lian's height is less than Hannah's height, you will need to subtract \begin{align*}x\end{align*} inches from Hannah's height to represent Lian's height.
\begin{align*}& \underline{x} \ inches \ \underline{shorter \ than} \ \underline{Hannah}.\\ & \Box \qquad \qquad \quad \downarrow \qquad \qquad \ \Box\\ & \Box \qquad \qquad \quad \downarrow \qquad \qquad \ \Box\\ & \Box \qquad \qquad \quad \downarrow \qquad \qquad \ \Box\\ & 65 \qquad \qquad \quad - \qquad \qquad x\end{align*}
The answer is \begin{align*}65-x\end{align*}.
If you knew what any of these expressions were equal to, then you could use an equals sign and write an equation. Take a look at these examples.
Write an equation for each phrase.
Example A
A number plus five is ten
Solution: \begin{align*}x + 5 = 10\end{align*}
Example B
Six more than a number is eighteen.
Solution:\begin{align*}x + 6 = 18\end{align*}
Example C
Fifteen less than a number is twenty
Solution:\begin{align*}x - 15 = 20\end{align*}
Here is the original problem once again.
Kara and her twin brother Marc are going to be spending one month in Boston with their grandparents. They are very excited! Not only is it summer in Boston and the weather will be terrific, but Boston has a fantastic subway system called the T and they will be able to ride it to get all around.
Kara is very excited about visiting the museums, but Marc who is a huge baseball fan is hoping for a trip to Fenway Park before the month is up. Seeing the Red Sox play would be a huge bonus!
Kara has earned $125.00 babysitting and has some birthday money too.
Before the trip, she counted her savings and realized that she has $54.25.
You can write an equation to show the unknown in this dilemma.
To write an equation for this dilemma, we can include what we know and what we don't know.
We know that Kara earned $125.00 babysitting.
We don't know how much money she received from her birthday. This is our unknown.
We do know the total amount of money.
Here is our equation.
\begin{align*}125 + x = 54.25\end{align*}
This is our answer.
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.
Write an equation to represent: Four less than an unknown number is eighteen.
Answer
To figure this out, we can first identify the operation as subtraction because of the key words less than.
Next, notice that the statement is four less than, so the four is being taken away from the unknown number.
Now we can write the equation.
\begin{align*}x - 4 = 18\end{align*}
This is our answer.
Video Review
Here is a video for review.
- This James Sousa video is an introduction to writing variable expressions.
Practice
Directions: 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
Directions: Write a simple equation for each phrase.
7. Five less than a number is ten.
8. Eight take away four is a number.
9. Five and a number is twelve.
10. Sixteen less than an unknown number is eighty.
11. Twenty and a number is fifty - five.
12. A number and fifteen is forty.
13. A number and twelve is sixty.
14. Fifteen less than a number is ninety.
15. Sixty less than a number is eighty.
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Image Attributions
Here you'll learn to write addition and subtraction phrases as single variable equations.