Algebraic fractions.

 

Simplify \[ \frac{y}{x} - \frac{x}{x + y} - \frac{xy}{x(x + y)} \ \]

Put everything over a common denominator.

\[ \begin{align} &= \frac{y(x + y) - x^2 - xy}{x(x + y)} \\ &= \frac{xy+ y^2 - x^2 - xy}{x(x + y)} \\ &= \frac{y^2 - x^2}{x(x + y)} \\ \end{align} \ \]

Factorise the numerator.

\[ \begin{align} &= \frac{(y - x)(y + x)}{x(x + y)} \\ \end{align}\ \]

Cancel common factors.

\[ \begin{align} &= \frac{y - x}{x} \\ \end{align} \ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

 

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Now You Try:

Simplify \[ \frac{1}{x-y} + \frac{2x-y}{x^2 - y^2} \ \]