Solve x+y/x-y+1+x-y/x+y

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-y}{x-y}.

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

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

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

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

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

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{2x}{x-y} times \frac{x+y}{x+y}. Multiply \frac{x-y}{x+y} times \frac{x-y}{x-y}.

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

Since \frac{2x\left(x+y\right)}{\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.

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-y}{x-y}.

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

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

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

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

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

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{2x}{x-y} times \frac{x+y}{x+y}. Multiply \frac{x-y}{x+y} times \frac{x-y}{x-y}.

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

Since \frac{2x\left(x+y\right)}{\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.

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

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

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

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

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

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

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