Solve 1-x/x-y/frac{y{x}-y/x-y}+x^2/y^2-xy

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

Evaluate

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

Short Solution Steps

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

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

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

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

Combine like terms in x-y-x.

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

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

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

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

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

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

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

Combine like terms in yx-y^{2}-yx.

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

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

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

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

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

Factor y^{2}-xy.

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

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

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

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

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

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

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

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

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

Cancel out y in both numerator and denominator.

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

Expand

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

Short Solution Steps

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

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

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

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

Combine like terms in x-y-x.

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

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

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

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

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

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

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

Combine like terms in yx-y^{2}-yx.

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

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

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

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

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

Factor y^{2}-xy.

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

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

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

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

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

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

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

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

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

Cancel out y in both numerator and denominator.

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