### Alex lives in Los Angeles and his best friend Gabriel lives 380 miles away in San Francisco. If he drives at a constant speed of 65 miles per hour all the way there, how long will it take Alex to get to San Francisco? (Assuming he doesn't stop.)

### Watch This

Khan Academy: Solving One-Step Equations

### Guidance

When solving an equation for a variable, you must get the variable *by itself.* All the equations in this lesson are **linear equations.** Here we will only deal with using one operation; addition, subtraction, multiplication or division.

### Example A

### Solve \begin{align*}7+y=16\end{align*} for \begin{align*}y\end{align*}.

**Solution:** This problem is simple and you could probably solve it in your head. However, to start good practices, you should always use algebra to solve any equation. Even if the problem seems easy, equations will get more difficult to solve.

To solve an equation for a variable, you must do the opposite, or undo, whatever is on the same side as the variable. 7 is being added to \begin{align*}y\end{align*}, so we must subtract 7 from both sides. Notice that this is very similar to the previous concept (*Solving Algebraic Equations for a Variable*).

\begin{align*}& \ \bcancel{7}+y = 16\\ & \underline{-\bcancel{7} \quad \quad -7 \; \;}\\ & \ \quad \ \ y = 9\end{align*}

You can check that \begin{align*}y= 9\end{align*} is correct by plugging 9 back into the original equation. 7 + 9 does equal 16, so we know that we found the correct answer.

#### Example B

### Solve \begin{align*}-7h=84\end{align*}.

**Solution:** Recall that \begin{align*}-7h = -7 \times h\end{align*}, so the opposite, or **inverse**, operation of multiplication is division. Therefore, we must divide both sides by -7 to solve for \begin{align*}h\end{align*}.

\begin{align*}\frac{-\bcancel{7}h}{-\bcancel{7}} &= \frac{84}{-7}\\ h &= -12\end{align*}

Again, check your work. \begin{align*}-7 \cdot -12\end{align*} is equal to 84, so we know our answer is correct.

#### Example C

### Solve \begin{align*}\frac{3}{8} x = \frac{3}{2}\end{align*}.

**Solution:** The variable is being multiplied by a fraction. Instead of dividing by a fraction, we multiply by the **reciprocal** of \begin{align*}\frac{3}{8}\end{align*}, which is \begin{align*}\frac{8}{3}\end{align*}.

\begin{align*}\xcancel{\frac{8}{3} \cdot \frac{3}{8}} x &= \bcancel{\frac{3}{2} \cdot \frac{8}{3}}\\ x &= \frac{8}{2}=4\end{align*}

Check the answer: \begin{align*}\frac{3}{_2\cancel{8}} \cdot \cancel{4}=\frac{3}{2}\end{align*}. This is correct, so we know that \begin{align*}x = 4\end{align*} is the answer.

###
**Intro Problem Revisited** --Set up an equation to represent Alex's travel, \begin{align*}t=d \div 65\end{align*} where *t* is time and *d* is distance. Therefore, it takes him \begin{align*}t=380 \div 65\end{align*} or 5.85 hours to get to Gabriel's house, which is 5 hours and 51 minutes.

### Guided Practice

Solve the following equations for the given variable. Check your answer.

1. \begin{align*}5+j=17\end{align*}

2. \begin{align*}\frac{h}{6}=-11\end{align*}

3. \begin{align*}\frac{5}{4}x=35\end{align*}

#### Answers

1. Subtract 5 from both sides to solve for \begin{align*}j\end{align*}.

\begin{align*}& \ \ \bcancel{5}+j=17\\ & \underline{-\bcancel{5} \qquad -5 \; \;}\\ & \qquad \ j=12\end{align*}

Check the answer: \begin{align*}5+12=17 \end{align*}

2. \begin{align*}h\end{align*} is being divided by 6. To undo division, we must multiply both sides by 6.

\begin{align*}\cancel{6} \cdot \frac{h}{\cancel{6}} &= -11.6\\ h &= -66\end{align*}

Check the answer: \begin{align*}\frac{-66}{6}=-11 \end{align*}

3. Multiply both sides by the reciprocal of \begin{align*}\frac{5}{4}\end{align*}

\begin{align*}\frac{\cancel{4}}{\cancel{5}} \cdot \frac{\cancel{5}}{\cancel{4}}x &= _7\cancel{35} \cdot \frac{4}{\cancel{5}}\\ x &=28\end{align*}

Check the answer: \begin{align*}\frac{5}{4} \cdot 28 = 5 \cdot 7 = 35\end{align*}

### Vocabulary

- Linear Equation
- An equation in one variable without exponents. Linear equations are in the form \begin{align*}ax \pm b=c\end{align*}, where \begin{align*}a, b\end{align*}, and \begin{align*}c\end{align*} are real numbers.

- Inverse
- The opposite operation of a given operation in an equation. For example, subtraction is the inverse of addition and multiplication is the inverse of division.

- Reciprocal
- The reciprocal of \begin{align*}\frac{a}{b}\end{align*} is \begin{align*}\frac{b}{a}\end{align*}.

### Practice

Solve each equation below and check your answer. Reduce any fractions.

- \begin{align*}-3+x=-1\end{align*}
- \begin{align*}r+6=2\end{align*}
- \begin{align*}5s=30\end{align*}
- \begin{align*}-8k=-64\end{align*}
- \begin{align*}\frac{m}{-4}=14\end{align*}
- \begin{align*}90=10n\end{align*}