Addition and subtraction of like fractions

Algebra: Adding and subtracting fractions

Addition and subtraction of like fractions

	Examples
When adding like fractions, the #\blue{\text{denominator }}# remains equal, and the #\orange{\text{numerators }}# are added.	\[\begin{array}{rcl} \dfrac{\orange{2x}}{\blue{y}} + \dfrac{\orange{x}}{\blue{y}} &=& \dfrac{\orange{3x}}{\blue{y}} \\ \end{array}\]

When subtracting like fractions, the #\blue{\text{denominator }}# remains equal, and the #\orange{\text{numerators}}# are subtracted.

\[\begin{array}{rcl}\dfrac{\orange{x}}{\blue{y}} - \dfrac{\orange{2x}}{\blue{y}} &=& \dfrac{\orange{-x}}{\blue{y}}
\end{array}\]

Simplifying

We always prefer to have fractions that are simplified as much as possible. Hence, after adding or subtracting fractions, we will check if the answer can be simplified further.

Examples

\[\begin{array}{rcl} &&\dfrac{\orange{x+5}}{\blue{x^2+6x+8}}+\dfrac{\orange{2x+7}}{\blue{x^2+6x+8}} \\&=&\dfrac{\orange{3x+12}}{\blue{x^2+6x+8}}\\ &=& \dfrac{\orange{3 \cdot (x+4)}}{\blue{(x+2) \cdot (x+4)}}\\&=&\dfrac{\orange{3}}{\blue{x+2}} \end{array}\]

Brackets

Please note, that when subtracting fractions, you should place brackets around the numerator. This it to make sure you are subtracting the entire numerator.

Examples

\[\begin{array}{rcl} \dfrac{\orange{x+5}}{\blue{x+6}} - \dfrac{\orange{x+3}}{\blue{x+6}} &=& \dfrac{\orange{x+5-(x+3)}}{\blue{x+6}}\\ &=& \dfrac{\orange{2}}{\blue{x+6}} \\ \\
\dfrac{\orange{x+5}}{\blue{x+6}} - \dfrac{\orange{x+3}}{\blue{x+6}} & \red\ne & \dfrac{\orange{x+5-x+3}}{\blue{x+6}}\\ &=& \dfrac{\orange{8}}{\blue{x+6}} \end{array}\]

Write as a single fraction and simplify as far as possible:
\[\dfrac{3\cdot a+8}{6-9\cdot a} - \dfrac{9-a}{6-9\cdot a}\]

# \dfrac{4\cdot a-1}{6-9\cdot a} #

#\begin{array}{rcl}
\dfrac{3\cdot a+8}{6-9\cdot a} - \dfrac{9-a}{6-9\cdot a} &=& \dfrac{3\cdot a+8 - \left(9-a\right)}{6-9\cdot a}\\
&& \phantom{xxx}\blue{\text{like fractions added by adding numerators}}\\
&=& \dfrac{4\cdot a-1}{6-9\cdot a} \\ && \phantom{xxx}\blue{\text{simplified}}\\
\end{array}#

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