Solve 1/x-y-x-y-1/x^2-2xy+y^2

$$\frac { 1 } { x - y } - \frac { x - y - 1 } { x ^ { 2 } - 2 x y + y ^ { 2 } }$$

Evaluate

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

Short Solution Steps

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-y and \left(x-y\right)^{2} is \left(x-y\right)\left(x-y\right)^{2}. Multiply \frac{1}{x-y} times \frac{\left(x-y\right)^{2}}{\left(x-y\right)^{2}}. Multiply \frac{x-y-1}{\left(x-y\right)^{2}} times \frac{x-y}{x-y}.

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

Since \frac{\left(x-y\right)^{2}}{\left(x-y\right)\left(x-y\right)^{2}} and \frac{\left(x-y-1\right)\left(x-y\right)}{\left(x-y\right)\left(x-y\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-y\right)^{2}-\left(x-y-1\right)\left(x-y\right).

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

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

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

Cancel out x-y in both numerator and denominator.

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

Expand \left(x-y\right)^{2}.

$$\frac{1}{x^{2}-2xy+y^{2}}$$

Expand

$$\frac{1}{x^{2}-2xy+y^{2}}$$

Short Solution Steps

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-y and \left(x-y\right)^{2} is \left(x-y\right)\left(x-y\right)^{2}. Multiply \frac{1}{x-y} times \frac{\left(x-y\right)^{2}}{\left(x-y\right)^{2}}. Multiply \frac{x-y-1}{\left(x-y\right)^{2}} times \frac{x-y}{x-y}.

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

Since \frac{\left(x-y\right)^{2}}{\left(x-y\right)\left(x-y\right)^{2}} and \frac{\left(x-y-1\right)\left(x-y\right)}{\left(x-y\right)\left(x-y\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-y\right)^{2}-\left(x-y-1\right)\left(x-y\right).

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

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

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

Cancel out x-y in both numerator and denominator.

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

Expand \left(x-y\right)^{2}.

$$\frac{1}{x^{2}-2xy+y^{2}}$$