Solve x^2-y^2/xy-1/x+y(x^2/y-y^2/x)]:frac{x-y}{x}

$$\frac { x ^ { 2 } - y ^ { 2 } } { x y } - \frac { 1 } { x + y } ( \frac { x ^ { 2 } } { y } - \frac { y ^ { 2 } } { x } ) ] : \frac { x - y } { x }$$

Evaluate

$$\frac{x-y}{x+y}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y and x is xy. Multiply \frac{x^{2}}{y} times \frac{x}{x}. Multiply \frac{y^{2}}{x} times \frac{y}{y}.

$$\frac{x^{2}-y^{2}}{xy}-\frac{1}{x+y}\left(\frac{x^{2}x}{xy}-\frac{y^{2}y}{xy}\right)$$

Since \frac{x^{2}x}{xy} and \frac{y^{2}y}{xy} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}-y^{2}}{xy}-\frac{1}{x+y}\times \frac{x^{2}x-y^{2}y}{xy}$$

Do the multiplications in x^{2}x-y^{2}y.

$$\frac{x^{2}-y^{2}}{xy}-\frac{1}{x+y}\times \frac{x^{3}-y^{3}}{xy}$$

Multiply \frac{1}{x+y} times \frac{x^{3}-y^{3}}{xy} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}-y^{2}}{xy}-\frac{x^{3}-y^{3}}{\left(x+y\right)xy}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of xy and \left(x+y\right)xy is xy\left(x+y\right). Multiply \frac{x^{2}-y^{2}}{xy} times \frac{x+y}{x+y}.

$$\frac{\left(x^{2}-y^{2}\right)\left(x+y\right)}{xy\left(x+y\right)}-\frac{x^{3}-y^{3}}{xy\left(x+y\right)}$$

Since \frac{\left(x^{2}-y^{2}\right)\left(x+y\right)}{xy\left(x+y\right)} and \frac{x^{3}-y^{3}}{xy\left(x+y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{2}-y^{2}\right)\left(x+y\right)-\left(x^{3}-y^{3}\right)}{xy\left(x+y\right)}$$

Do the multiplications in \left(x^{2}-y^{2}\right)\left(x+y\right)-\left(x^{3}-y^{3}\right).

$$\frac{x^{3}+x^{2}y-y^{2}x-y^{3}-x^{3}+y^{3}}{xy\left(x+y\right)}$$

Combine like terms in x^{3}+x^{2}y-y^{2}x-y^{3}-x^{3}+y^{3}.

$$\frac{-y^{2}x+x^{2}y}{xy\left(x+y\right)}$$

Factor the expressions that are not already factored in \frac{-y^{2}x+x^{2}y}{xy\left(x+y\right)}.

$$\frac{xy\left(x-y\right)}{xy\left(x+y\right)}$$

Cancel out xy in both numerator and denominator.

$$\frac{x-y}{x+y}$$

Expand

$$\frac{x-y}{x+y}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y and x is xy. Multiply \frac{x^{2}}{y} times \frac{x}{x}. Multiply \frac{y^{2}}{x} times \frac{y}{y}.

$$\frac{x^{2}-y^{2}}{xy}-\frac{1}{x+y}\left(\frac{x^{2}x}{xy}-\frac{y^{2}y}{xy}\right)$$

Since \frac{x^{2}x}{xy} and \frac{y^{2}y}{xy} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}-y^{2}}{xy}-\frac{1}{x+y}\times \frac{x^{2}x-y^{2}y}{xy}$$

Do the multiplications in x^{2}x-y^{2}y.

$$\frac{x^{2}-y^{2}}{xy}-\frac{1}{x+y}\times \frac{x^{3}-y^{3}}{xy}$$

Multiply \frac{1}{x+y} times \frac{x^{3}-y^{3}}{xy} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{2}-y^{2}}{xy}-\frac{x^{3}-y^{3}}{\left(x+y\right)xy}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of xy and \left(x+y\right)xy is xy\left(x+y\right). Multiply \frac{x^{2}-y^{2}}{xy} times \frac{x+y}{x+y}.

$$\frac{\left(x^{2}-y^{2}\right)\left(x+y\right)}{xy\left(x+y\right)}-\frac{x^{3}-y^{3}}{xy\left(x+y\right)}$$

Since \frac{\left(x^{2}-y^{2}\right)\left(x+y\right)}{xy\left(x+y\right)} and \frac{x^{3}-y^{3}}{xy\left(x+y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{2}-y^{2}\right)\left(x+y\right)-\left(x^{3}-y^{3}\right)}{xy\left(x+y\right)}$$

Do the multiplications in \left(x^{2}-y^{2}\right)\left(x+y\right)-\left(x^{3}-y^{3}\right).

$$\frac{x^{3}+x^{2}y-y^{2}x-y^{3}-x^{3}+y^{3}}{xy\left(x+y\right)}$$

Combine like terms in x^{3}+x^{2}y-y^{2}x-y^{3}-x^{3}+y^{3}.

$$\frac{-y^{2}x+x^{2}y}{xy\left(x+y\right)}$$

Factor the expressions that are not already factored in \frac{-y^{2}x+x^{2}y}{xy\left(x+y\right)}.

$$\frac{xy\left(x-y\right)}{xy\left(x+y\right)}$$

Cancel out xy in both numerator and denominator.

$$\frac{x-y}{x+y}$$