Solve (x/x-y-2y/x-y)xy/x-2ydiv(1/x+frac{1}{y})

$$( \frac { x } { x - y } - \frac { 2 y } { x - y } ) \frac { x y } { x - 2 y } \div ( \frac { 1 } { x } + \frac { 1 } { y } )$$

Evaluate

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

Short Solution Steps

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

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

Multiply \frac{x-2y}{x-y} times \frac{xy}{x-2y} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x-2y in both numerator and denominator.

$$\frac{\frac{xy}{x-y}}{\frac{1}{x}+\frac{1}{y}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and y is xy. Multiply \frac{1}{x} times \frac{y}{y}. Multiply \frac{1}{y} times \frac{x}{x}.

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

Since \frac{y}{xy} and \frac{x}{xy} have the same denominator, add them by adding their numerators.

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

Divide \frac{xy}{x-y} by \frac{y+x}{xy} by multiplying \frac{xy}{x-y} by the reciprocal of \frac{y+x}{xy}.

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

Multiply x and x to get x^{2}.

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

Multiply y and y to get y^{2}.

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

Consider \left(x-y\right)\left(y+x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{x^{2}y^{2}}{x^{2}-y^{2}}$$

Expand

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

Short Solution Steps

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

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

Multiply \frac{x-2y}{x-y} times \frac{xy}{x-2y} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x-2y in both numerator and denominator.

$$\frac{\frac{xy}{x-y}}{\frac{1}{x}+\frac{1}{y}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and y is xy. Multiply \frac{1}{x} times \frac{y}{y}. Multiply \frac{1}{y} times \frac{x}{x}.

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

Since \frac{y}{xy} and \frac{x}{xy} have the same denominator, add them by adding their numerators.

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

Divide \frac{xy}{x-y} by \frac{y+x}{xy} by multiplying \frac{xy}{x-y} by the reciprocal of \frac{y+x}{xy}.

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

Multiply x and x to get x^{2}.

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

Multiply y and y to get y^{2}.

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

Consider \left(x-y\right)\left(y+x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{x^{2}y^{2}}{x^{2}-y^{2}}$$