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# Single Variable Subtraction Equations

## Solve one - step equations using addition.

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Single Variable Subtraction Equations
Credit: Yellowstone National Park

Steve and his family are going on a mini-vacation or a weekend getaway to the beach. Steve and his two sisters are excited about staying at a hotel on the beach. When they arrive at the hotel, Steve’s father, Mr. Richards, goes to the registration desk to check for his reservation. The manager tells him it will cost 288 after he receives a 10% discount on the two-room family suite. Mr. Richards thanks the manager for the discount. How can Mr. Richard figure out how much the suite was before the discount? In this concept, you will learn to solve single-variable subtraction equations. ### Solving Single-Variable Subtraction Equations You can use inverses to solve single-variable subtraction equations. The goal is to get the variable alone on one side of the equal signs and using the inverse of subtraction – addition, can help you do this. Keep in mind that what you do to one side of the equation you have to do to the other side. Here is an equation. \begin{align*}x - 12 = 40\end{align*} To solve this equation, you can use the inverse of subtraction (addition) and add 12 to both sides of the equation. That will help to get the variable alone and solve the problem. First, identify the number being subtracted from the variable. -12 Next, using the inverse of 12, add 12 to both sides of the equation. \begin{align*}\begin{array}{rcl} && x-12\ = \ \ 40\\ && \underline{\quad + 12 \quad \ +12}\\ && \ x-0 \ \ = \ \ 52 \end{array}\end{align*} The plus 12 cancels the minus 12 on the left side of the equation, leaving only \begin{align*}x\end{align*}. On the other side of the equation, 12 and 40 are added to get 52. The answer is \begin{align*}x = 52\end{align*}. To check this answer, substitute it back into the original problem and see if the statement is true. \begin{align*}\begin{array}{rcl} x-12 &=& 40\\ 52-12 &=& 40\\ 40 &=& 40 \end{array}\end{align*} The answer is true. Sometimes, in single-variable subtraction equations, the variable will be subtracted from a number. Take a look at the equation below. \begin{align*}32 - x = 7\end{align*} One way to solve this equation without getting into positive and negative numbers is to simply turn it into a single-variable addition equation by using the inverse of subtraction and adding \begin{align*}x\end{align*} to both sides. First, since the variable \begin{align*}x\end{align*} is being subtracted, add \begin{align*}x\end{align*} to both sides. \begin{align*}\begin{array}{rcl} && 32 - x = 7\\ && \ \underline{\;\;\;\;\; +x = 7 + x}\\ && \ \quad \ \ 32 = 7 + x \end{array}\end{align*} Now, you have a single-variable addition equation. \begin{align*}x + 7 = 32\end{align*} Next, solve the addition equation by using the inverse of addition and subtracting 7 from both sides. \begin{align*}\begin{array}{rcl} && x + 7 = 32\\ && \underline{\;\;\;\; - 7 = -7}\\ && \qquad x = 25 \end{array}\end{align*} The answer is \begin{align*}x = 25\end{align*}. Then, check your answer by substituting 25 for \begin{align*}x\end{align*} in the original equation. \begin{align*}\begin{array}{rcl} 32 - x & = & 7\\ 32 - 25 & = & 7\\ 7 & = & 7 \end{array}\end{align*} The answer checks out. ### Examples #### Example 1 Earlier, you were given a problem about Steve and his family’s weekend beach getaway. Mr. Richards wants to know the usual cost of the room if he paid288 after a 32 discount. First, write an equation for the situation. The total cost (\begin{align*}x\end{align*}) minus 32 (discount) is 288. Or \begin{align*}x - 32 = 288\end{align*} Next, identify the number being subtracted from the variable. - 32 Then, using the inverse of -32, add 32 to both sides of the equation. \begin{align*}\begin{array}{rcl} && x - 32 = 288\\ && \underline{\;\;\;\; +32 \ \ +32}\\ && \ \ x - 0 = 320 \end{array}\end{align*} The answer is \begin{align*}x = \320\end{align*}. To check this answer, substitute320 back into the original problem and see if the statement is true.

\begin{align*}\begin{array}{rcl} x -32 &=& 288\\ 320 - 32 &=& 288\\ 320 &=& 320 \end{array}\end{align*}

The room cost \$320 before the discount.

#### Example 2

Solve the following equation.

\begin{align*}y- 21 = 59\end{align*}

First, identify the number being subtracted from the variable.

-21

Next, using the inverse of 21, add 21 to both sides of the equation.

\begin{align*}\begin{array}{rcl} && y - 21 = 59\\ && \underline{\;\;\;\; +21 \ \ + 21}\\ && \ \ y - 0 = 80 \end{array}\end{align*}

The answer is \begin{align*}y = 80\end{align*}.

To check this answer, substitute it back into the original problem and see if the statement is true.

\begin{align*}\begin{array}{rcl} y - 21 &=& 59\\ 80 - 21 &=& 59\\ 59 &=& 59 \end{array}\end{align*}

Solve each equation and write your answer in the format: \begin{align*}x = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}.

#### Example 3

\begin{align*}x-9 = 22\end{align*}

First, identify the number being subtracted from the variable.

-9

Next, using the inverse of 9, add 9 to both sides of the equation.

\begin{align*}\begin{array}{rcl} && \ \ x - 9 = 22\\ &&\ \ \underline{ \ \ +9 \ \ \ +9}\\ && \ \ x - 0 = 31 \end{array}\end{align*}

The answer is \begin{align*}x = 31\end{align*}.

To check this answer, substitute it back into the original problem and see if the statement is true.

\begin{align*}\begin{array}{rcl} x - 9 &=& 22\\ 31 - 9 &=& 22\\ 22 &=& 22 \end{array}\end{align*}

#### Example 4

\begin{align*}x-3 = 46\end{align*}

First, identify the number being subtracted from the variable.

-3

Next, using the inverse of -3, add 3 to both sides of the equation.

\begin{align*}\begin{array}{rcl} && \ \ x - 3 = 46\\ && \ \ \underline{\ \ +3 \ \ \ +3}\\ && \ \ x - 0 = 49 \end{array}\end{align*}

The answer is \begin{align*}x = 49\end{align*}.

To check this answer, substitute it back into the original problem and see if the statement is true.

\begin{align*}\begin{array}{rcl} x -3 &=& 46\\ 49 - 3 &=& 46\\ 46 &=& 46 \end{array}\end{align*}

#### Example 5

\begin{align*}x- 7 = 23\end{align*}

First, identify the number being subtracted from the variable.

-7

Next, using the inverse of -7, add 7 to both sides of the equation.

\begin{align*}\begin{array}{rcl} && \ \ x - 7 = 23\\ &&\ \ \underline{ \ \ +7 \ \ \ +7}\\ && \ \ x - 0 = 30 \end{array}\end{align*}

The answer is \begin{align*}x = 30\end{align*}.

To check this answer, substitute it back into the original problem and see if the statement is true.

\begin{align*}\begin{array}{rcl} x - 7 &=& 23\\ 30 - 7 &=& 23\\ 23 &=& 23 \end{array}\end{align*}

### Review

Solve each single-variable subtraction problem using the inverse operation. Write your answer in the form: \begin{align*}\text{variable} = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}.

1. \begin{align*}y-5=10\end{align*}
2. \begin{align*}x-7=17\end{align*}
3. \begin{align*}a-4=12\end{align*}
4. \begin{align*}z-6=22\end{align*}
5. \begin{align*}y-9=11\end{align*}
6. \begin{align*}b-5=12\end{align*}
7. \begin{align*}x-8=30\end{align*}
8. \begin{align*}y-7=2\end{align*}
9. \begin{align*}x-9=1\end{align*}
10. \begin{align*}x-19=15\end{align*}
11. \begin{align*}x-18=12\end{align*}
12. \begin{align*}x-29=31\end{align*}
13. \begin{align*}x-15=62\end{align*}
14. \begin{align*}x-22=45\end{align*}
15. \begin{align*}x-19=37\end{align*}

To see the Review answers, open this PDF file and look for section 12.6.

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### Vocabulary Language: English

TermDefinition
Difference The result of a subtraction operation is called a difference.
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.
Simplify To simplify means to rewrite an expression to make it as "simple" as possible. You can simplify by removing parentheses, combining like terms, or reducing fractions.
Sum The sum is the result after two or more amounts have been added together.