Solving for a Variable:

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

Remember back to the lesson on fractions where we learned about multiplicative inverses of fractions.
Multiplicative Inverse: Number when multiplied to another number equals $1$

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

Video Source (10:05 mins) | Transcript

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

$\frac{2}{5}\rightarrow \frac{5}{2}$
$-\frac{1}{3}\rightarrow -3$
$\frac{7}{8}\rightarrow \frac{8}{7}$

Additional Resources

Khan Academy: One-step Multiplication Equations (02:22 mins, Transcript)
Khan Academy: One-step Multiplication & Division Equations: Fractions & Decimals (07:50 mins, Transcript)
Khan Academy: Linear Equations 1 (07:27 mins, Transcript)

Practice Problems

Solve for the variable:

$-7{\text{M}}=\frac{-7}{4}$

$\frac{6}{5}{\text{B}}=3$

$\frac{-2}{3}{\text{g}}=-1$

$\frac{-2}{7}=\frac{-3}{2}{\text{x}}$

$4{\text{j}}=\frac{3}{2}$

$\frac{-3}{5}=\frac{3}{2}{\text{D}}$

$\frac{-{\text{J}}}{3}=\frac{-7}{6}$

Solutions

$\frac{1}{4}$ (Written Solution)

In this example, we want to get the variable ${\text{M}}$ all by itself on one side of the equation. We do this by unraveling the equation using the order of operations backward.

The only operation on the same side of the equation as ${\text{M}}$ is multiplication.

We can solve for ${\text{M}}$ by multiplying both sides of the equation by the multiplicative inverse of $-7$.

$(-7)(\frac{-1}{7})=1$ so we will multiply both sides by $\frac{-1}{7}$.

$This is a picture of the equation -7M=\frac{-7}{4}. There is a vertical dashed line through the equal sign. “Left-hand side” is written on the left of the dashed line, with “right-hand side” written on the right. On the second line (\frac{-1}{7})(-7M)=(\frac{-1}{7})(\frac{-7}{4}) is written.$

The left side of the equation:

$\frac{-1}{7}$ multiplied to $-7$ is $1$ since they are the multiplicative inverses of each other.

The right side of the equation:

Multiply across numerators and denominators to get $\frac{7}{28}$.

 $This is a picture of the equation (\frac{-1}{7})(-7M)=(\frac{-1}{7})(\frac{-7}{4}). There is a vertical dashed line through the equal sign. “Left-hand side” is written on the left of the dashed line, with “right-hand side” written on the right. On the second line \frac{(-1)\times(-7)}{7 \times 1}(M)=\frac{(-1)\times(-7)}{7 \times 4} is written. The next line is: \frac{7}{7}(M)=\frac{7}{28}. The next line is M=\frac{7}{28}.$

The fraction $\frac{7}{28}$ simplifies to $\frac{1}{4}$.

${\text{M}}={\color{Blue} \frac{7}{28}=\frac{7 \times 1}{7 \times 4}=\frac{{\color{Red} \cancel{7}}\times 1}{{\color{Red} \cancel{7}} \times 4}=\frac{1}{4}}$

Our final solution: ${\text{M}} = \frac{1}{4}$

$\frac{5}{2}$

$\frac{3}{2}$ (Solution Video | Transcript)

$\frac{4}{21}$ (Written Solution)

In this example, we want to solve for the variable ${\text{X}}$. It doesn’t matter if it is on the right or left side of the equation. We do the exact same process.

In order to isolate ${\text{X}}$ we need to multiply both sides by the multiplicative inverse of $(\frac{-3}{2})$.

$(\frac{-3}{2})(-\frac{2}{3})= 1$ so we will multiply both sides by $(-\frac{2}{3})$.

$This is a picture of the equation \frac{-2}{7}=\frac{-3}{2}x. There is a vertical dashed line through the equal sign. “Left-hand side” is written on the left of the dashed line, with “right-hand side” written on the right. On the second line (\frac{-2}{3})\frac{-2}{7}=(\frac{-2}{3})\frac{-3}{2}x is written.$

Remember, it doesn’t matter if the negative is in the numerator, denominator, or just out in front of the fraction. As long as there is only one negative sign, the entire fraction is negative.

Now we multiply across numerators and denominators to simplify both sides of the equation.

$This is a picture of the equation (\frac{-2}{3})\frac{-2}{7}=(\frac{-2}{3})\frac{-3}{2}x. There is a vertical dashed line through the equal sign. “Left-hand side” is written on the left of the dashed line, with “right-hand side” written on the right. On the second line \frac{4}{21}=\frac{6}{6}x is written. The last line is \frac{4}{21}=1x.$

In this example, our variable ${\text{X}}$ is on the right side of the equal sign. It doesn’t matter which side it is on as long as it is all by itself.

$\frac{4}{21}={\text{X}}$

Therefore: ${\text{X}} = \frac{4}{21}$

$\frac{4}{21}$ can not be simplified.

Our final solution: ${\text{X}} = \frac{4}{21}$

$\frac{3}{8}$

$\frac{-2}{5}$

$\frac{7}{2}$ (Solution Video | Transcript)