**Introduction**

In this lesson, you will practice necessary skills for solving for a variable on one side of an equation using multiplication, division, and fractions.

This video illustrates the lesson material below. Watching the video is optional.

- Solving for a Variable on One Side: Part 5 - Multiplication and Division with Fractions (10:05 mins) | Transcript

**Solving for a Variable on One Side Using Multiplication and Division with Fractions**

Review the multiplicative inverses of fractions:

**Multiplicative Inverse**: Number when multiplied to another number equals 1.

When solving equations where the variable is being multiplied by a fraction, follow the same steps as when it was a whole number and multiply both sides by the multiplicative inverse.

The multiplicative inverse is the reciprocal of the original fraction, but the sign stays the same:

- \( {\dfrac {2} {5}}\) → \( {\dfrac {5} {2}}\)
- \( {-\dfrac {1}{3}}\) → \(−3\)
- \( {\dfrac {7} {8}}\) → \( {\dfrac {8} {7}} \)

**Example 1**

Solve for \(x\): \(\frac{2}{5}x = 8\)

Fractions are the same as division. \(\frac{2}{5}x=8\) is the same as \((2\div5)x=8\). In order to solve this equation, isolate \(x\). This means \(x\) needs to be alone on the left-hand side.

If you want to isolate \(x\), you need to remove the \(\frac{2}{5}\) by turning it into 1. Do this by multiplying both sides of the equation by the multiplicative inverse of \(\frac{2}{5}\), which is \(\frac{5}{2}\). Solve this equation using the multiplicative inverse:

\begin{align*}\frac{2}{5}x &=8 &\color{red}\small\text{Simplify each side}\\\\ \frac{2}{5}x \color{red}\mathbf{(\frac{5}{2})} &= \frac{8}{1} \color{red}\mathbf{(\frac{5}{2})} &\color{red}\small\text{Multiplicative inverse of \(\frac{2}{5}\) is \(\frac{5}{2}\)}\\\\ x &= \frac{8}{1} \color{red}\mathbf{(\frac{5}{2})} &\color{red}\small\text{ Left side: \(\frac{2}{5}\) and \(\frac{5}{2}\) will cancel out}\\\\ x &= \frac{40}{2} &\color{red}\small\text{Multiply}\\\\x&=20 &\color{red}\small\text{Simplify}\\\\\end{align*}

Remember, you can find the multiplicative inverse of a fraction by switching the locations of the numbers in the numerator and the denominator. \(\frac{2}{5}\cdot \frac{5}{2} = 1\)

To check the answer, substitute 20 for \(x\) in the original equation of \(\frac{2}{5}x=8\).

\begin{align*}

\frac{2}{5}x &= 8 &\color{red}\small\text{Check the value of x}\\\\

\frac{2}{5}(20) &= 8 &\color{red}\small\text{Substitute 20 in for x}\\\\

\frac{2}{5}\cdot \frac{20}{1} &= 8 &\color{red}\small\text{Rewrite 20 as \(\frac{20}{1}\)}\\\\

\frac{40}{5} &= 8 &\color{red}\small\text{Multiply}\\\\

8&=8 &\color{red}\small\text{Simplify}\\\\\\\\

\end{align*}

Since the left side is equal to the right side of the equation (\( 8 = 8\)) and makes this statement true, \(x\) is equal to 20.

In the previous example, you saw an equation that was being multiplied by a fraction. A fraction inherently has division in it. How would you solve something like Example 2?

**Example 2**

\begin{align*}\frac{x}{3} &= 6 &\color{red}\small\text{Solve for the value of x}\\\\\frac{1}{3}x &= 6 &\color{red}\small\text{Left side: \(\frac{x}{3}\) is the same as \(\frac{1}{3}x\)}\\\\\frac{1}{3}x\color{red}\mathbf{(\frac{3}{1})} &= \frac{6}{1} \color{red}\mathbf{(\frac{3}{1})} &\color{red}\small\text{Multiplicative inverse of \(\frac{1}{3}\) is \(\frac{3}{1}\)}\\\\ x &= \frac{6}{1} \color{red}\mathbf{(\frac{3}{1})} &\color{red}\small\text{Left side: \(\frac{1}{3}\) and \(\frac{3}{1}\) will cancel out}\\\\ x &= \frac{18}{1} &\color{red}\small\text{Multiply}\\\\x &= 18 &\color{red}\small\text{Simplify}\\\\\end{align*}

To check the answer, substiute 18 for \(x\).

\begin{align*} \frac{x}{3} &= 6 &\color{red}\small\text{Check the value of x}\\\\ \frac{18}{3} &= 6 &\color{red}\small\text{Substitute 18 in for x}\\\\ 6 &= 6 &\color{red}\small\text{Simplify} \end{align*}

This is true. Also, \(18\div3=6\), so \(x=18\) is the correct solution for this equation.

**Things to Remember**

- Fractions are a type of division.
- Multiply both sides of the equation by the multiplicative inverse of a coefficient to isolate the variable.

### Practice Problems

**Solve for the variable:**

- \(-7{\text{M}}=-\dfrac{7}{4}\) (Solution
- \(\dfrac{6}{5}{\text{B}}=3\) (Solution
- \(-\dfrac{2}{3}{\text{g}}=-1\) (Video Solution
- \(-\dfrac{2}{7}=-\dfrac{3}{2}{\text{x}}\) (Solution
- \(4{\text{j}}=\dfrac{3}{2}\) (Solution
- \(-\dfrac{3}{5}=\dfrac{3}{2}{\text{D}}\) (Solution
- \(-\dfrac{{\text{J}}}{3}=-\dfrac{7}{6}\) (Video Solution

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