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?

### Linear Equations

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*}, where \begin{align*}a, b\end{align*}, and \begin{align*}c\end{align*} are real numbers. Here we will only deal with using one operation; addition, subtraction, multiplication or division.

Let's solve each equation for the specified variable.

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

This problem is simple and you could probably solve it in your head. However, for good practice, 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.

\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.

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

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.

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

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.

### Examples

#### Example 1

Earlier, you were asked to find the time it will take Alex to get from Los Angeles to San Francisco.

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.

#### Example 2

Solve the following equation for the given variable: \begin{align*}5+j=17\end{align*}. Check your answer.

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*}

#### Example 3

Solve the following equation for the given variable: \begin{align*}\frac{h}{6}=-11\end{align*}. Check your answer.

\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*}

#### Example 4

Solve the following equation for the given variable: \begin{align*}\frac{5}{4}x=35\end{align*}. Check your answer.

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*}

### Review

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 equations 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.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 1.7.