Solve [(1/x-1/y):(1/x+1/y)+frac{x-y}{x+y}]:(frac{1}{x}+frac{1}{x+y})=

$$[ ( \frac { 1 } { x } - \frac { 1 } { y } ) : ( \frac { 1 } { x } + \frac { 1 } { y } ) + \frac { x - y } { x + y } ] : ( \frac { 1 } { x } + \frac { 1 } { x + y } ) =$$

Evaluate

$$0$$

Short Solution Steps

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}.

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

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

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

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

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

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

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

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

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

Cancel out xy in both numerator and denominator.

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

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

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

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

$$\frac{\frac{0}{x+y}}{\frac{1}{x}+\frac{1}{x+y}}$$

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

$$\frac{\frac{0}{x+y}}{\frac{x+y}{x\left(x+y\right)}+\frac{x}{x\left(x+y\right)}}$$

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

$$\frac{\frac{0}{x+y}}{\frac{x+y+x}{x\left(x+y\right)}}$$

Combine like terms in x+y+x.

$$\frac{\frac{0}{x+y}}{\frac{2x+y}{x\left(x+y\right)}}$$

Divide \frac{0}{x+y} by \frac{2x+y}{x\left(x+y\right)} by multiplying \frac{0}{x+y} by the reciprocal of \frac{2x+y}{x\left(x+y\right)}.

$$\frac{0x\left(x+y\right)}{\left(x+y\right)\left(2x+y\right)}$$

Cancel out x+y in both numerator and denominator.

$$\frac{0}{2x+y}$$

Zero divided by any non-zero term gives zero.

$$0$$

Factor

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