Solving for a Variable:

Solving for a Variable on One Side Using Addition and Subtraction with Fractions

When solving for variables with fractions instead of whole numbers, we still use the same process: adding the additive inverse.
Additive inverse: Number when added to another number equals $0$.
The following video will show what the additive inverse of a fraction is and review some addition and subtraction of fractions:

Video Source (09:59 mins) | Transcript

As a rule, we can remember additive inverses as having the opposite sign (positive or negative) as our number. This is true with fractions as well.
Also remember, in order to add or subtract fractions, they must have a common denominator.

Additional Resources

Khan Academy: One-step Addition & Subtraction: Fractions & Decimals (04:31 mins, Transcript)

Practice Problems

Solve for the variable:

$\frac{-9}{8}+{\text{g}}=\frac{-3}{8}$

${\text{r}}+\frac{1}{4}=-1$

${\text{x}}+\frac{5}{9}=\frac{1}{3}$

$-3+{\text{g}}=\frac{5}{2}$

$-4={\text{j}}+\frac{4}{3}$

$\frac{7}{6}={\text{A}}{-}\frac{1}{2}$

$\frac{-9}{8}=\frac{-3}{4}+{\text{a}}$

Solutions

$\frac{3}{4}$ (Written Solution)

In this example, we are solving for the variable ${\text{g}}$. The only operation involved between ${\text{g}}$ and other numbers is addition. We solve this the same way we solve equations with integers. We unravel the equation using the order of operations backward.

Step 1: Do the inverse of any addition or subtraction

On the left-hand side of the equal sign $-\frac{9}{8}$ is being added to ${\text{g}}$. We can remove this by adding the inverse to both sides of the equation.

Add $+\frac{9}{8}$ to both sides.

The left side of the equation:

$-\frac{9}{8}+\frac{9}{8}=0$ leaving just the variable ${\text{g}}$

The right side of the equation:

Since both of these fractions have the same denominator we can add the numerators.

$-\frac{3}{8}+\frac{9}{8}=\frac{6}{8}$

$This is a picture of the equation \frac{-9}{8}+g=-⅜. There is a vertical dashed line through the equal sign. “Left-hand side” is written on the left of the dashed line, with “right-hand side” written on the right. \frac{+9}{8} is written below \frac{-9}{8} and below -⅜. There is a horizontal line beneath that indicating that addition has been performed. Below that g=\frac{6}{8} has been written.$

Normally this would be the end of our work since we isolated the solution for variable ${\text{g}}$, but in this case, we can still simplify our fraction.

$6$ and $8$ both have $2$ as a common factor.

$6=(2)( 3)$

$8=(2)(4)$

We can divide out the $2{\text{’s}}$ leaving just $\frac{3}{4}$.

${\text{g}}=\frac{6}{8}=\frac{\left ( 2 \right )\left ( 3 \right )}{\left ( 2 \right )\left ( 4 \right )}=\frac{{\color{Red} \cancel{\left ( 2 \right )} }\left ( 3 \right )}{{\color{Red} \cancel{\left ( 2 \right )}}\left ( 4 \right )}=\frac{3}{4}$

Our final solution: ${\text{g}} = \frac{3}{4}$

$\frac{-5}{4}$

$-\frac{2}{9}$ (Written Solution)

In this example, we need to get the variable ${\text{X}}$ all alone. To do this we add the inverse of $\frac{5}{9}$ to both sides of the equation.

We add $-\frac{5}{9}$ to both sides.

$This is a picture of the equation x+\frac{5}{9}=\frac{1}{3}. There is a vertical dashed line through the equal sign. “Left-hand side” is written on the left of the dashed line, with “right-hand side” written on the right. On the second line x+\frac{5}{9}+ -\frac{5}{9}=\frac{1}{3}+ -\frac{5}{9} is written.$

The left side of the equation:

Positive $\frac{5}{9}$ plus negative $\frac{5}{9}$ equals $0$. This leaves just the variable ${\text{X}}$ by itself.

$This is a picture of the equation x+\frac{5}{9}+ -\frac{5}{9}=\frac{1}{3}+ -\frac{5}{9}. There is a vertical dashed line through the equal sign. “Left-hand side” is written on the left of the dashed line, with “right-hand side” written on the right. On the second line x+ 0=\frac{1}{3}+ -\frac{5}{9} is written. There is a horizontal bracket indicating that \frac{5}{9}+ -\frac{5}{9} is equal to 0.$

The right side of the equation:

In order to add $\frac{1}{3}+-\frac{5}{9}$ we need to first get common denominators.

$9$ is a multiple of $3$, so our least common multiple is $9$.

Multiply $\frac{1}{3}$ by $1$ in the form of $\frac{3}{3}$ to change its denominator to $9$.

${\text{X}}=\left (\frac{1}{3} \right ){\color{Cyan} \left (\frac{3}{3} \right )}{\color{DarkGreen} + {\color{Red} -}\frac{5}{9}}$

${\text{X}}=\frac{1 \cdot {\color{Cyan} 3}}{3 \cdot {\color{Cyan} 3}}{\color{DarkGreen} + {\color{Red} -}\frac{5}{9}}$

${\text{X}}={\color{Cyan} \frac{3}{9}} {\color{DarkGreen} + {\color{Red} -}\frac{5}{9}}$

Now that our fractions have the same denominator, we can add the numerators.

$3+-5=-2$

${\text{X}}=\frac{{\color{Cyan} 3}}{9} + {\color{Red} -}\frac{{\color{DarkGreen} 5}}{9}$

${\text{X}}=\frac{{\color{Cyan} 3+{\color{Red} -}{\color{DarkGreen} 5}}}{9}$

${\text{X}}=\frac{{\color{Red} -}{\color{Orange} 2}}{9}$

Our final solution: ${\text{X}} = -\frac{2}{9}$

$\frac{11}{2}$

$\frac{-16}{3}$ (Solution Video | Transcript)

$\frac{5}{3}$

$\frac{-3}{8}$ (Solution Video | Transcript)