# 2.1: Equations with Variables on One Side

**Advanced**Created by: CK-12

**Practice**One-Step Equations and Inverse Operations

Erin, Jillian, Stephanie and Jacob went to the movies. The total bill for the tickets and snacks came to $72.00. What is an equation that represents this situation? How much should each teen pay to split the bill evenly?

### Equations with Variables on One Side

When solving any equation, your job is to find the value for the letter that makes the equation true. Solving equations with variables on one side can be done with the help of models such as a balance or algebra tiles.

When solving equations with variables on one side of the equation there is one main rule to follow: whatever you do to one side of the equals sign you must do the same to the other side of the equals sign. For example, if you add a number to the left side of an equals sign, you must add the same number to the right side of the equals sign.

#### Let's practice by solving an equation by solving for the variable in the following problems:

- \begin{align*}5a + 2 = 17\end{align*}

The problem can be solved if you think about the problem in terms of a balance. You know that the two sides are equal so the balance has to stay horizontal. You can place each side of the equation on each side of the balance.

In order to solve the equation, you have to get the variable \begin{align*}a\end{align*} all by itself. Always remember that you need to keep the balance horizontal. This means that whatever you do to one side of the equation, you have to do to the other side.

First subtract 2 from both sides to get rid of the 2 on the left.

Since 5 is multiplied by \begin{align*}a\end{align*}, you can get \begin{align*}a\end{align*} by itself (or isolate it) by dividing by 5. Remember that whatever you do to one side, you have to do to the other.

If you simplify this expression, you get:

Therefore \begin{align*}a = 3\end{align*}.

You can check your answer to see if you are correct by substituting your answer back into the original equation.

\begin{align*}5a + 2 &= 17\\ 5({\color{red}3}) + 2 &= 17\\ 15 + 2 &= 17\\ 17 &= 17 \ \ \end{align*}

- \begin{align*}7b - 7 = 42\end{align*}

Again, you can solve the problem if you think about the problem in terms of a balance (or a seesaw). You know that the two sides are equal so the balance has to stay horizontal. You can place each side of the equation on each side of the balance.

In order to solve the equation, you have to get the variable \begin{align*}b\end{align*} all by itself. Always remember that you need to keep the balance horizontal. This means that whatever you do to one side of the equation, you have to do to the other side.

First add 7 from both sides to get rid of the 7 on the left.

Since 7 is multiplied by \begin{align*}b\end{align*}, you can get \begin{align*}b\end{align*} by itself (or isolate it) by dividing by 7. Remember that whatever you do to one side, you have to do to the other.

If you simplify this expression, you get:

Therefore \begin{align*}b = 7\end{align*}.

You can check your answer to see if you are correct.

\begin{align*}7b - 7 &= 42\\ 7({\color{red}7}) - 7 &= 42\\ 49 - 7 &= 42\\ 42 &= 42 \ \ \end{align*}

#### Now let's practice solving solving an equation using algebra tiles:

This same method can be extended by using algebra tiles. If you let rectangular tiles represent the variable, square tiles represent one unit, green tiles represent positives numbers, and white tiles represent the negative numbers, you can solve equations using an alternate method.

Take the equation \begin{align*}3c + 2 = 11\end{align*}.

The green algebra \begin{align*}x-\end{align*}tiles represent variables; therefore, there are 3 \begin{align*}c\end{align*} blocks for the equation. The other green blocks represent the numbers or constants. There is a 2 on the left side of the equation so there are 2 square green blocks. There is an 11 on the right side of the equation so there are 11 square green blocks on the right side of the equation.

To solve, add two negative tiles to the right and left hand sides. The same rule applies to this problem as to all of the previous problems. Whatever you do to one side you have to do to the other.

This leaves us with the following:

You can reorganize these to look like the following:

Organizing the remaining algebra tiles allows us to realize the answer to be \begin{align*}x = 3\end{align*} or for your example \begin{align*}c = 3\end{align*}.

Let’s do your check as with the previous two problems.

\begin{align*}3c + 2 &= 11\\ 3({\color{red}3}) + 2 &= 11\\ 9 + 2 &= 11\\ 11 &= 11 \ \ \end{align*}

### Examples

#### Example 1

Earlier, you were told that Erin, Jillian, Stephanie, and Jacob went to the movies. The total bill was $72.00. What is an equation that represents this situation? How much should each teen pay to split the bill evenly?

Since there were four teens going to the movies (Erin, Jillian, Stephanie, and Jacob) and the total is $72.00, your equation is \begin{align*}4x = 72\end{align*}. you divide by 4 to find your answer.

\begin{align*}x &= \frac{72}{4}\\ x &= 18\end{align*}

Therefore each teen will have to pay $18.00 for their movie ticket and snack.

#### Example 2

Use a model to solve for the variable in the equation \begin{align*}x-5=12\end{align*}.

\begin{align*}x-5=12\end{align*}

Therefore, \begin{align*}x=17\end{align*}.

#### Example 3

Use a different model to solve for the variable in the equation \begin{align*}3y+9=12\end{align*}.

\begin{align*}3y+9=12\end{align*}

First you have to subtract 9 from both sides of the equation in order to start to isolate the variable.

Now, in order to get \begin{align*}y\end{align*} all by itself, you have to divide both sides by 3. This will isolate the variable \begin{align*}y\end{align*}.

Therefore, \begin{align*}y=1\end{align*}.

#### Example 4

Solve for \begin{align*}x\end{align*} in the equation \begin{align*}3x-2x+16=-3\end{align*}.

\begin{align*}3x-2x+16=-3\end{align*}

You can use any method to solve this equation. Remember to isolate the \begin{align*}x\end{align*} variable. You will notice here that there are two \begin{align*}x\end{align*} values on the left. First let’s combine these terms.

\begin{align*}3x-2x+16 &= -3\\ x+16 &= -3\end{align*}

Now you can use any method to solve the equation. You now should just have to subtract 16 from both sides to isolate the \begin{align*}x\end{align*} variable.

\begin{align*}x+16 {\color{red}-16} &= -3 {\color{red}-16}\\ x &= -19\end{align*}

### Review

Use the models that you have learned to solve for the variables in the following problems.

- \begin{align*}a+3=-5\end{align*}
- \begin{align*}2b-1=5\end{align*}
- \begin{align*}4c-3=9\end{align*}
- \begin{align*}2-d=3\end{align*}
- \begin{align*}4-3e=-2\end{align*}
- \begin{align*}x+3=14\end{align*}
- \begin{align*}2y-7=5\end{align*}
- \begin{align*}3z+6=9\end{align*}
- \begin{align*}5+3x=-3\end{align*}
- \begin{align*}2x+2=-4\end{align*}
- \begin{align*}-4x+13=5\end{align*}
- \begin{align*}3x-5=22\end{align*}
- \begin{align*}11-2x=5\end{align*}
- \begin{align*}2x-4=4\end{align*}
- \begin{align*}5x+3=28\end{align*}

For each of the following models, write a problem involving an equation with a variable on one side of the equation expressed by the model and then solve for the variable.

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### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.1.

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Term | Definition |
---|---|

constant |
A constant is a value that does not change. In Algebra, this is a number such as 3, 12, 342, etc., as opposed to a variable such as x, y or a. |

Equation |
An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs. |

Numerical Coefficient |
In mathematical expressions, the numerical coefficients are the numbers associated with the variables. For example, in the expression , 4 is the numerical coefficient and is the variable. |

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. |

### Image Attributions

Here you will begin your study of mathematical equations by learning how to solve equations with a variable on only one side.