Solve 1/x^{2}+2/xy-frac{x^{2}+y^{2}}{x^2y^2}-3/y^2+4x/xy^2

$$\frac { 1 } { x ^ { 2 } } + \frac { 2 } { x y } - \frac { x ^ { 2 } + y ^ { 2 } } { x ^ { 2 } y ^ { 2 } } - \frac { 3 } { y ^ { 2 } } + \frac { 4 x } { x y ^ { 2 } }$$

Evaluate

$$\frac{2}{xy}$$

Short Solution Steps

Cancel out x in both numerator and denominator.

$$\frac{1}{x^{2}}+\frac{2}{xy}-\frac{x^{2}+y^{2}}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{y}{yx^{2}}+\frac{2x}{yx^{2}}-\frac{x^{2}+y^{2}}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{y+2x}{yx^{2}}-\frac{x^{2}+y^{2}}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{\left(y+2x\right)y}{x^{2}y^{2}}-\frac{x^{2}+y^{2}}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{\left(y+2x\right)y-\left(x^{2}+y^{2}\right)}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{y^{2}+2xy-x^{2}-y^{2}}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{2xy-x^{2}}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{x\left(-x+2y\right)}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

Cancel out x in both numerator and denominator.

$$\frac{-x+2y}{xy^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of xy^{2} and y^{2} is xy^{2}. Multiply \frac{3}{y^{2}} times \frac{x}{x}.

$$\frac{-x+2y}{xy^{2}}-\frac{3x}{xy^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{-x+2y-3x}{xy^{2}}+\frac{4}{y^{2}}$$

Combine like terms in -x+2y-3x.

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

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

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

Since \frac{-4x+2y}{xy^{2}} and \frac{4x}{xy^{2}} have the same denominator, add them by adding their numerators.

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

Combine like terms in -4x+2y+4x.

$$\frac{2y}{xy^{2}}$$

Cancel out y in both numerator and denominator.

$$\frac{2}{xy}$$

Expand

$$\frac{2}{xy}$$

Short Solution Steps

Cancel out x in both numerator and denominator.

$$\frac{1}{x^{2}}+\frac{2}{xy}-\frac{x^{2}+y^{2}}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{y}{yx^{2}}+\frac{2x}{yx^{2}}-\frac{x^{2}+y^{2}}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{y+2x}{yx^{2}}-\frac{x^{2}+y^{2}}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{\left(y+2x\right)y}{x^{2}y^{2}}-\frac{x^{2}+y^{2}}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{\left(y+2x\right)y-\left(x^{2}+y^{2}\right)}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{y^{2}+2xy-x^{2}-y^{2}}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{2xy-x^{2}}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{x\left(-x+2y\right)}{x^{2}y^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

Cancel out x in both numerator and denominator.

$$\frac{-x+2y}{xy^{2}}-\frac{3}{y^{2}}+\frac{4}{y^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of xy^{2} and y^{2} is xy^{2}. Multiply \frac{3}{y^{2}} times \frac{x}{x}.

$$\frac{-x+2y}{xy^{2}}-\frac{3x}{xy^{2}}+\frac{4}{y^{2}}$$

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

$$\frac{-x+2y-3x}{xy^{2}}+\frac{4}{y^{2}}$$

Combine like terms in -x+2y-3x.

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

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

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

Since \frac{-4x+2y}{xy^{2}} and \frac{4x}{xy^{2}} have the same denominator, add them by adding their numerators.

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

Combine like terms in -4x+2y+4x.

$$\frac{2y}{xy^{2}}$$

Cancel out y in both numerator and denominator.

$$\frac{2}{xy}$$