Solve 4/x^2-1/y^2/frac{2{x}+1/y}

$$\frac { \frac { 4 } { x ^ { 2 } } - \frac { 1 } { y ^ { 2 } } } { \frac { 2 } { x } + \frac { 1 } { y } }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and y^{2} is x^{2}y^{2}. Multiply \frac{4}{x^{2}} times \frac{y^{2}}{y^{2}}. Multiply \frac{1}{y^{2}} times \frac{x^{2}}{x^{2}}.

$$\frac{\frac{4y^{2}}{x^{2}y^{2}}-\frac{x^{2}}{x^{2}y^{2}}}{\frac{2}{x}+\frac{1}{y}}$$

Since \frac{4y^{2}}{x^{2}y^{2}} and \frac{x^{2}}{x^{2}y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{4y^{2}-x^{2}}{x^{2}y^{2}}}{\frac{2}{x}+\frac{1}{y}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and y is xy. Multiply \frac{2}{x} times \frac{y}{y}. Multiply \frac{1}{y} times \frac{x}{x}.

$$\frac{\frac{4y^{2}-x^{2}}{x^{2}y^{2}}}{\frac{2y}{xy}+\frac{x}{xy}}$$

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

$$\frac{\frac{4y^{2}-x^{2}}{x^{2}y^{2}}}{\frac{2y+x}{xy}}$$

Divide \frac{4y^{2}-x^{2}}{x^{2}y^{2}} by \frac{2y+x}{xy} by multiplying \frac{4y^{2}-x^{2}}{x^{2}y^{2}} by the reciprocal of \frac{2y+x}{xy}.

$$\frac{\left(4y^{2}-x^{2}\right)xy}{x^{2}y^{2}\left(2y+x\right)}$$

Cancel out xy in both numerator and denominator.

$$\frac{-x^{2}+4y^{2}}{xy\left(x+2y\right)}$$

Factor the expressions that are not already factored.

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

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

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and y^{2} is x^{2}y^{2}. Multiply \frac{4}{x^{2}} times \frac{y^{2}}{y^{2}}. Multiply \frac{1}{y^{2}} times \frac{x^{2}}{x^{2}}.

$$\frac{\frac{4y^{2}}{x^{2}y^{2}}-\frac{x^{2}}{x^{2}y^{2}}}{\frac{2}{x}+\frac{1}{y}}$$

Since \frac{4y^{2}}{x^{2}y^{2}} and \frac{x^{2}}{x^{2}y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{4y^{2}-x^{2}}{x^{2}y^{2}}}{\frac{2}{x}+\frac{1}{y}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and y is xy. Multiply \frac{2}{x} times \frac{y}{y}. Multiply \frac{1}{y} times \frac{x}{x}.

$$\frac{\frac{4y^{2}-x^{2}}{x^{2}y^{2}}}{\frac{2y}{xy}+\frac{x}{xy}}$$

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

$$\frac{\frac{4y^{2}-x^{2}}{x^{2}y^{2}}}{\frac{2y+x}{xy}}$$

Divide \frac{4y^{2}-x^{2}}{x^{2}y^{2}} by \frac{2y+x}{xy} by multiplying \frac{4y^{2}-x^{2}}{x^{2}y^{2}} by the reciprocal of \frac{2y+x}{xy}.

$$\frac{\left(4y^{2}-x^{2}\right)xy}{x^{2}y^{2}\left(2y+x\right)}$$

Cancel out xy in both numerator and denominator.

$$\frac{-x^{2}+4y^{2}}{xy\left(x+2y\right)}$$

Factor the expressions that are not already factored.

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

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

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