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# 3.2: One-Step Equations Transformed by Multiplication-Division

Difficulty Level: At Grade Created by: CK-12
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Practice One-Step Equations Transformed by Multiplication/Division

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What if you had an algebraic equation involving multiplication or division like 5x=3\begin{align*}-5x = 3\end{align*}? How could you solve it for the unknown variable x? After completing this Concept, you'll be able to solve equations like this one.

### Guidance

#### Example C

Solve 5x=3.25\begin{align*}5x = 3.25\end{align*}.

To cancel the 5, we divide both sides by 5.

5x5x=3.255=0.65\begin{align*} \frac{5x}{5} &= \frac{3.25}{5}\\ x &= 0.65\end{align*}

#### Example D

Solve 1.375x=1.2\begin{align*}1.375x = 1.2\end{align*}.

Divide by 1.375

xx=1.21.375=0.872¯¯¯¯¯\begin{align*} x &= \frac{1.2}{1.375}\\ x &= 0.8 \overline{72}\end{align*}

Notice the bar above the final two decimals; it means that those digits recur, or repeat. The full answer is 0.872727272727272....

To see more examples of one - and two-step equation solving, watch the Khan Academy video series starting at http://www.youtube.com/watch?v=bAerID24QJ0.

Watch this video for help with the Examples above.

### Vocabulary

• An equation in which each term is either a constant or the product of a constant and a single variable is a linear equation.
• We can add, subtract, multiply, or divide both sides of an equation by the same value and still have an equivalent equation.
• To solve an equation, isolate the unknown variable on one side of the equation by applying one or more arithmetic operations to both sides.

### Guided Practice

Solve:

a) 9x5=5\begin{align*} \frac{9x}{5} = 5\end{align*}.

b) 7x=511\begin{align*}7x = \frac{5}{11}\end{align*}.

Solutions:

a) 9x5\begin{align*}\frac{9x}{5}\end{align*} is equivalent to 95x\begin{align*} \frac{9}{5} \cdot x\end{align*}, so to cancel out that 95\begin{align*} \frac{9}{5}\end{align*}, we multiply by the reciprocal, 59\begin{align*} \frac{5}{9}\end{align*}.

59(9x5)x=59(5)=259\begin{align*} \frac{5}{9} \left ( \frac{9x}{5} \right ) &= \frac{5}{9}(5)\\ x &= \frac{25}{9}\end{align*}

b) Divide both sides by 7.

xx=511.7=577\begin{align*}x &= \frac{5}{11.7}\\ x &= \frac{5}{77} \end{align*}

### Practice

For 1-5, solve the following equations for x\begin{align*}x\end{align*}.

1. 7x=21\begin{align*}7x = 21 \end{align*}
2. 4x=1\begin{align*}4x = 1 \end{align*}
3. 5x12=23\begin{align*}\frac{5x}{12} = \frac{2}{3}\end{align*}
4. 0.01x=11\begin{align*}0.01x = 11\end{align*}
5. 2x9=103\begin{align*}\frac{-2x}{9} = \frac{10}{3}\end{align*}

For 6-10, solve the following equations for the unknown variable.

1. 21s=3\begin{align*}21s = 3 \end{align*}
2. 7a=5\begin{align*}-7a = -5 \end{align*}
3. 7f11=711\begin{align*}\frac{7f}{11} = \frac{7}{11} \end{align*}
4. 6r=38\begin{align*}6r = \frac{3}{8} \end{align*}
5. 9b16=38\begin{align*}\frac{9b}{16} = \frac{3}{8} \end{align*}

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