Solve x^2/y^2-1/frac{x^2{y^2}-2x/y+1}

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{y^{2}}{y^{2}}.

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{y^{2}}{y^{2}}.

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

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

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

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

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

Cancel out y^{2} in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x-y in both numerator and denominator.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{y^{2}}{y^{2}}.

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{y^{2}}{y^{2}}.

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

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

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

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

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

Cancel out y^{2} in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x-y in both numerator and denominator.

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