Solve (1/x1/y)*(x-y)+(1/x-1/y)*(x+y)

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

Evaluate

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

Short Solution Steps

Multiply \frac{1}{x} times \frac{1}{y} by multiplying numerator times numerator and denominator times denominator.

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

Express \frac{1}{xy}\left(x-y\right) as a single fraction.

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

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}.

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

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

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

Express \frac{y-x}{xy}\left(x+y\right) as a single fraction.

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

Since \frac{x-y}{xy} and \frac{\left(y-x\right)\left(x+y\right)}{xy} have the same denominator, add them by adding their numerators.

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

Do the multiplications in x-y+\left(y-x\right)\left(x+y\right).

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

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

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

Expand

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

Short Solution Steps

Multiply \frac{1}{x} times \frac{1}{y} by multiplying numerator times numerator and denominator times denominator.

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

Express \frac{1}{xy}\left(x-y\right) as a single fraction.

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

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}.

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

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

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

Express \frac{y-x}{xy}\left(x+y\right) as a single fraction.

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

Since \frac{x-y}{xy} and \frac{\left(y-x\right)\left(x+y\right)}{xy} have the same denominator, add them by adding their numerators.

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

Do the multiplications in x-y+\left(y-x\right)\left(x+y\right).

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

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

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