Solve 1/x+y+1/x-y/frac{1{x+y}-1/x-y}=

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+y and x-y is \left(x+y\right)\left(x-y\right). Multiply \frac{1}{x+y} times \frac{x-y}{x-y}. Multiply \frac{1}{x-y} times \frac{x+y}{x+y}.

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+y and x-y is \left(x+y\right)\left(x-y\right). Multiply \frac{1}{x+y} times \frac{x-y}{x-y}. Multiply \frac{1}{x-y} times \frac{x+y}{x+y}.

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

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

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

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

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

Combine like terms in x-y-x-y.

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

Divide \frac{2x}{\left(x+y\right)\left(x-y\right)} by \frac{-2y}{\left(x+y\right)\left(x-y\right)} by multiplying \frac{2x}{\left(x+y\right)\left(x-y\right)} by the reciprocal of \frac{-2y}{\left(x+y\right)\left(x-y\right)}.

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

Cancel out 2\left(x+y\right)\left(x-y\right) in both numerator and denominator.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+y and x-y is \left(x+y\right)\left(x-y\right). Multiply \frac{1}{x+y} times \frac{x-y}{x-y}. Multiply \frac{1}{x-y} times \frac{x+y}{x+y}.

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+y and x-y is \left(x+y\right)\left(x-y\right). Multiply \frac{1}{x+y} times \frac{x-y}{x-y}. Multiply \frac{1}{x-y} times \frac{x+y}{x+y}.

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

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

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

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

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

Combine like terms in x-y-x-y.

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

Divide \frac{2x}{\left(x+y\right)\left(x-y\right)} by \frac{-2y}{\left(x+y\right)\left(x-y\right)} by multiplying \frac{2x}{\left(x+y\right)\left(x-y\right)} by the reciprocal of \frac{-2y}{\left(x+y\right)\left(x-y\right)}.

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

Cancel out 2\left(x+y\right)\left(x-y\right) in both numerator and denominator.

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