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.** That means the equation can be simplified to \begin{align*}ax+b=c\end{align*}

#### Example A

Solve \begin{align*}7+y=16\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*}*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*}

#### Example B

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

**Solution:** Recall that \begin{align*}-7h = -7 \times h\end{align*}**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*}

#### 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*}

\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*}

**Intro Problem Revisit** Set up an equation to represent Alex's travel, \begin{align*}t=d \div 65\end{align*}*t* is time and *d* is distance. Therefore, it takes him \begin{align*}t=380 \div 65\end{align*}

### 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*}

\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*}

### Explore More

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*}
- \begin{align*}-16=y-5\end{align*}
- \begin{align*}\frac{6}{7}d=36\end{align*}
- \begin{align*}6=-\frac{1}{3}p\end{align*}
- \begin{align*}u-\frac{3}{4}=\frac{5}{6}\end{align*}
- \begin{align*}\frac{8}{5}a=-\frac{72}{13}\end{align*}
- \begin{align*}\frac{7}{8}=b+\frac{1}{2}\end{align*}
- \begin{align*}w-(-5)=16\end{align*}
- \begin{align*}\frac{1}{4}=b-\left(-\frac{2}{5}\right)\end{align*}
- \begin{align*}\frac{3}{5}q=-\frac{12}{11}\end{align*}
- \begin{align*}\frac{t}{12}=-4\end{align*}
- \begin{align*}45=15x\end{align*}
- \begin{align*}7=\frac{g}{-8}\end{align*}

**Challenge** Solve the equation below. Be careful!

- \begin{align*}14-z=-3\end{align*}
- Alex decides he'd rather drive 70 miles an hour to get up to San Francisco. How long will it take him to drive up north now? Remember it was 380 miles to get to San Francisco from Los Angeles.