Adding and Subtracting Fractions With Variables
To add or subtract fractions:
- You must have a common denominator.
- To find the Least Common Denominator (LCD), take the least common multiple of the individual denominators.
- Express each fraction as a new fraction with the common denominator, by multiplying by one in an appropriate form.
-
To add fractions with the same denominator:
add the numerators,
and keep the denominator the same.
That is, use the rule:
$$ \frac{A}{C} + \frac{B}{C} = \frac{A+B}{C} $$
Example
Question:
Combine into a single fraction:
$$
\cssId{s12}{\frac{2}{x+3} - \frac{3x}{x-1}}
$$
Solution:
Note that the LCD is $\,(x+3)(x-1)\,.$
| $\displaystyle\frac{2}{x+3} - \frac{3x}{x-1}$ | original expression |
|
$\displaystyle
= \frac{2}{x+3}\cdot\frac{x-1}{x-1}
$
$\displaystyle \qquad -\ \frac{3x}{x-1}\cdot\frac{x+3}{x+3} $ |
get a common denominator by multiplying by $\,1\,$ |
| $\displaystyle = \frac{2(x-1)-3x(x+3)}{(x+3)(x-1)}$ | keep the denominator the same; add the numerators |
| $\displaystyle = \frac{2x-2-3x^2 - 9x}{(x+3)(x-1)}$ | multiply out the numerator |
| $\displaystyle = \frac{-3x^2 - 7x - 2}{(x+3)(x-1)}$ | combine like terms; write numerator in standard form |
Leave the denominator in factored form for your final answer.