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Adding and subtracting algebraic fractions

Recall how to add and subtract fractions.

When the denominators of two fractions are the same, we used the common denominator and add the numerators. For example:


When the denominators of the fractions to be added are different, we first find equivalent fractions with the lowest common denominator. For example:

\begin{align*} \dfrac{3}{7}+\dfrac{2}{5}&=\dfrac{15}{35}+\dfrac{14}{35}\\\\ &=\dfrac{29}{35} \end{align*} or \begin{align*} \dfrac{3}{7}+\dfrac{2}{5}&=\dfrac{15+14}{35}\\\\ &=\dfrac{29}{35} \end{align*}

Subtraction is undertaken in a similar manner.

With pronumerals, the second notation is most useful.

Example 7

Express as a single fraction Solution
\(\dfrac{x}{7}+\dfrac{4x}{7}\) \begin{align*} \dfrac{x}{7}+\dfrac{4x}{7} &=\dfrac{x+4x}{7} \\\\ &=\dfrac{5x}{7} \end{align*}
\(\dfrac{2z}{11}-\dfrac{z}{2}\) \begin{align*} \dfrac{2z}{11}-\dfrac{z}{2} &=\dfrac{4z-11z}{22} \\\\ &=\dfrac{-7z}{22}\\\\ &=-\dfrac{7z}{22} \end{align*}
\(-\dfrac{m}{3}+\dfrac{2m}{5}\) \begin{align*} -\dfrac{m}{3}+\dfrac{2m}{5} &=\dfrac{-5m+6m}{15} \\\\ &=\dfrac{m}{15} \end{align*}