Solve 2xy-y^2/x^2-y^2+x-y/x+y

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

Evaluate

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

Short Solution Steps

Factor x^{2}-y^{2}.

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

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

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

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

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

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

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

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

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

Expand \left(x+y\right)\left(x-y\right).

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

Expand

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

Short Solution Steps

Factor x^{2}-y^{2}.

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

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

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

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

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

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

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

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

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

Expand \left(x+y\right)\left(x-y\right).

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