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

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

Evaluate

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

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}{y} times \frac{x}{x}. Multiply \frac{y}{x} times \frac{y}{y}.

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

Since \frac{xx}{xy} and \frac{yy}{xy} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in xx-yy.

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

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}{y} times \frac{x}{x}. Multiply \frac{y}{x} times \frac{y}{y}.

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

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

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

Do the multiplications in xx+yy.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{xy}{xy}.

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

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

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

Divide \frac{x^{2}-y^{2}}{xy} by \frac{x^{2}+y^{2}-2xy}{xy} by multiplying \frac{x^{2}-y^{2}}{xy} by the reciprocal of \frac{x^{2}+y^{2}-2xy}{xy}.

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

Cancel out xy in both numerator and denominator.

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

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

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

Cancel out x-y in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

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

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

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

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

Cancel out x+y in both numerator and denominator.

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

Expand

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

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}{y} times \frac{x}{x}. Multiply \frac{y}{x} times \frac{y}{y}.

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

Since \frac{xx}{xy} and \frac{yy}{xy} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in xx-yy.

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

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}{y} times \frac{x}{x}. Multiply \frac{y}{x} times \frac{y}{y}.

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

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

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

Do the multiplications in xx+yy.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{xy}{xy}.

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

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

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

Divide \frac{x^{2}-y^{2}}{xy} by \frac{x^{2}+y^{2}-2xy}{xy} by multiplying \frac{x^{2}-y^{2}}{xy} by the reciprocal of \frac{x^{2}+y^{2}-2xy}{xy}.

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

Cancel out xy in both numerator and denominator.

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

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

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

Cancel out x-y in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

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

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

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

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

Cancel out x+y in both numerator and denominator.

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