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.)
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Khan Academy: Solving OneStep 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*}7+y=16 for \begin{align*}y\end{align*}y .
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*}
\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*}−7h=84 .
Solution: Recall that \begin{align*}7h = 7 \times 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*}38x=32 .
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 Revisited Set up an equation to represent Alex's travel, \begin{align*}t=d \div 65\end{align*}t=d÷65 where t is time and d is distance. Therefore, it takes him \begin{align*}t=380 \div 65\end{align*}t=380÷65 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*}
\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*}
ax±b=c , where \begin{align*}a, b\end{align*}a,b , and \begin{align*}c\end{align*}c 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*}
ab is \begin{align*}\frac{b}{a}\end{align*}ba .
Practice
Solve each equation below and check your answer. Reduce any fractions.

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