What if you were in a math contest and were given the equation \begin{align*}x+4=16\end{align*}? Or how about the equation \begin{align*}9x = 72\end{align*}? Could you solve each of these equations in one step? After completing this Concept, you'll be able to solve equations like these in a just a single step by using inverse operations.

### One-Step Equations and Inverse Operations

**It’s Easier than You Think**

You have been solving equations since the beginning of this textbook, although you may not have recognized it. For example, in a previous Concept, you determined the answer to the pizza problem below.

$20.00 *was one-quarter of the money spent on pizza.*

\begin{align*}\frac{1}{4} m=20.00\end{align*} What divided by 4 equals 20.00?

The solution is 80. So, the amount of money spent on pizza was $80.00.

By working through this question mentally, you were applying mathematical rules and solving for the variable \begin{align*}m\end{align*}.

**Definition:** To **solve** an equation means to write an equivalent equation that has the variable by itself on one side. This is also known as **isolating the variable.**

In order to begin solving equations, you must understand three basic concepts of algebra: inverse operations, equivalent equations, and the Addition Property of Equality.

**Inverse Operations and Equivalent Equations**

In another Concept, you learned how to simplify an expression using the Order of Operations: **P**arentheses, **E**xponents, **M**ultiplication and **D**ivision completed in order from left to right, and **A**ddition and **S**ubtraction (also completed from left to right). Each of these operations has an **inverse.** Inverse operations “undo” each other when combined.

For example, the inverse of addition is subtraction. The inverse of an exponent is a root.

**Definition: Equivalent equations** are two or more equations having the same solution.

**The Addition Property of Equality**

Just like Spanish, chemistry, or even music, mathematics has a set of rules you must follow in order to be successful. These rules are called properties, theorems, or axioms. They have been proven or agreed upon years ago, so you can apply them to many different situations.

For example, the **Addition Property of Equality** allows you to apply the same operation to each side of an equation, or “what you do to one side of an equation you can do to the other.”

*The Addition Property of Equality*

For all real numbers \begin{align*}a, b,\end{align*} and \begin{align*}c\end{align*}:

If \begin{align*}a = b\end{align*}, then \begin{align*}a + c = b + c\end{align*}.

**Solving One-Step Equations Using Addition or Subtraction**

Because subtraction can be considered “adding a negative,” the Addition Property of Equality also works if you need to subtract the same value from each side of an equation.

#### Solve Using Addition Property of Equality

*Solve for* \begin{align*}y\end{align*}*:*

\begin{align*}16 = y-11\end{align*}.

When asked to solve for \begin{align*}y\end{align*}, your goal is to write an equivalent equation with the variable \begin{align*}y\end{align*} isolated on one side.

Write the original equation: \begin{align*}16 = y-11\end{align*}.

Apply the Addition Property of Equality: \begin{align*}16 + 11 = y - 11 + 11\end{align*}.

Simplify by adding like terms: \begin{align*}27 = y\end{align*}.

The solution is \begin{align*}y = 27\end{align*}.

#### Solve Using the Addition Property of Equality

*Solve for* \begin{align*}z:\end{align*}

\begin{align*}5=z+12\end{align*}

Apply the Addition Property of Equality:

\begin{align*}&5=z+12\\ &5-12=z+12-12\\ &5-12=z\\ &-7=z \end{align*}

The solution is \begin{align*}-7=z.\end{align*}

Equations that take one step to isolate the variable are called **one-step equations**. Such equations can also involve multiplication or division.

**Solving One-Step Equations Using Multiplication or Division**

**The Multiplication Property of Equality**

For all real numbers \begin{align*}a, b\end{align*}, and \begin{align*}c\end{align*}:

*If* \begin{align*}a = b\end{align*}, *then* \begin{align*}a(c)= b(c).\end{align*}

#### Solve using the Multiplication Property of Equality

*Solve for* \begin{align*}k: -8k= -96.\end{align*}

Because \begin{align*}-8k= -8 \times k\end{align*}, the inverse operation of multiplication is division. Therefore, we must cancel multiplication by applying the Multiplication Property of Equality.

Write the original equation: \begin{align*}-8k= -96\end{align*}.

Apply the Multiplication Property of Equality: \begin{align*}-8k \div -8 = -96 \div -8.\end{align*}

The solution is \begin{align*}k= 12\end{align*}.

When working with fractions, you must remember: \begin{align*}\frac{a}{b} \times \frac{b}{a} = 1\end{align*}. In other words, in order to cancel a fraction using division, you really must multiply by its reciprocal.

### Examples

#### Example 1

*Determine the inverse of division.*

To undo division by a number, you would multiply by the same number.

#### Example 2

*Solve* \begin{align*}\frac{1}{8} \cdot x = 1.5\end{align*}.

The variable \begin{align*}x\end{align*} is being multiplied by one-eighth. Instead of dividing two fractions, we multiply by the reciprocal of \begin{align*}\frac{1}{8}\end{align*}, which is 8.

\begin{align*}\cancel{8} \left (\frac{1}{\cancel{8}} \cdot x \right ) & = 8(1.5) \\ x & = 12\end{align*}

### Review

Solve for the given variable.

- \begin{align*}x + 11 = 7\end{align*}
- \begin{align*}x - 1.1 = 3.2\end{align*}
- \begin{align*}7x = 21\end{align*}
- \begin{align*}4x = 1\end{align*}
- \begin{align*}\frac{5x}{12} = \frac{2}{3}\end{align*}
- \begin{align*}x + \frac{5}{2} = \frac{2}{3}\end{align*}
- \begin{align*}x - \frac{5}{6} = \frac{3}{8}\end{align*}
- \begin{align*}0.01x = 11\end{align*}
- \begin{align*}q - 13 = -13\end{align*}
- \begin{align*}z + 1.1 = 3.0001\end{align*}
- \begin{align*}21s = 3\end{align*}
- \begin{align*}t + \frac{1}{2} = \frac{1}{3}\end{align*}
- \begin{align*}\frac{7f}{11} = \frac{7}{11}\end{align*}
- \begin{align*}\frac{3}{4} = - \frac{1}{2} \cdot y\end{align*}
- \begin{align*}6r = \frac{3}{8}\end{align*}
- \begin{align*}\frac{9b}{16} = \frac{3}{8}\end{align*}

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 3.1.