Solve 5xy-5x/2y^{2+y-3}*4y+6/5x^{2-10xy+5y^2}divx^2-xy/x^2-2xy+y^2

$$\frac { 5 x y - 5 x } { 2 y ^ { 2 } + y - 3 } \times \frac { 4 y + 6 } { 5 x ^ { 2 } - 10 x y + 5 y ^ { 2 } } \div \frac { x ^ { 2 } - x y } { x ^ { 2 } - 2 x y + y ^ { 2 } }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{5xy-5x}{2y^{2}+y-3}.

$$\frac{\frac{5x\left(y-1\right)}{\left(y-1\right)\left(2y+3\right)}\times \frac{4y+6}{5x^{2}-10xy+5y^{2}}}{\frac{x^{2}-xy}{x^{2}-2xy+y^{2}}}$$

Cancel out y-1 in both numerator and denominator.

$$\frac{\frac{5x}{2y+3}\times \frac{4y+6}{5x^{2}-10xy+5y^{2}}}{\frac{x^{2}-xy}{x^{2}-2xy+y^{2}}}$$

Multiply \frac{5x}{2y+3} times \frac{4y+6}{5x^{2}-10xy+5y^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{5x\left(4y+6\right)}{\left(2y+3\right)\left(5x^{2}-10xy+5y^{2}\right)}}{\frac{x^{2}-xy}{x^{2}-2xy+y^{2}}}$$

Factor the expressions that are not already factored in \frac{x^{2}-xy}{x^{2}-2xy+y^{2}}.

$$\frac{\frac{5x\left(4y+6\right)}{\left(2y+3\right)\left(5x^{2}-10xy+5y^{2}\right)}}{\frac{x\left(x-y\right)}{\left(x-y\right)^{2}}}$$

Cancel out x-y in both numerator and denominator.

$$\frac{\frac{5x\left(4y+6\right)}{\left(2y+3\right)\left(5x^{2}-10xy+5y^{2}\right)}}{\frac{x}{x-y}}$$

Divide \frac{5x\left(4y+6\right)}{\left(2y+3\right)\left(5x^{2}-10xy+5y^{2}\right)} by \frac{x}{x-y} by multiplying \frac{5x\left(4y+6\right)}{\left(2y+3\right)\left(5x^{2}-10xy+5y^{2}\right)} by the reciprocal of \frac{x}{x-y}.

$$\frac{5x\left(4y+6\right)\left(x-y\right)}{\left(2y+3\right)\left(5x^{2}-10xy+5y^{2}\right)x}$$

Cancel out x in both numerator and denominator.

$$\frac{5\left(4y+6\right)\left(x-y\right)}{\left(2y+3\right)\left(5x^{2}-10xy+5y^{2}\right)}$$

Factor the expressions that are not already factored.

$$\frac{2\times 5\left(2y+3\right)\left(x-y\right)}{5\left(2y+3\right)\left(x-y\right)^{2}}$$

Cancel out 5\left(2y+3\right)\left(x-y\right) in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{5xy-5x}{2y^{2}+y-3}.

$$\frac{\frac{5x\left(y-1\right)}{\left(y-1\right)\left(2y+3\right)}\times \frac{4y+6}{5x^{2}-10xy+5y^{2}}}{\frac{x^{2}-xy}{x^{2}-2xy+y^{2}}}$$

Cancel out y-1 in both numerator and denominator.

$$\frac{\frac{5x}{2y+3}\times \frac{4y+6}{5x^{2}-10xy+5y^{2}}}{\frac{x^{2}-xy}{x^{2}-2xy+y^{2}}}$$

Multiply \frac{5x}{2y+3} times \frac{4y+6}{5x^{2}-10xy+5y^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{5x\left(4y+6\right)}{\left(2y+3\right)\left(5x^{2}-10xy+5y^{2}\right)}}{\frac{x^{2}-xy}{x^{2}-2xy+y^{2}}}$$

Factor the expressions that are not already factored in \frac{x^{2}-xy}{x^{2}-2xy+y^{2}}.

$$\frac{\frac{5x\left(4y+6\right)}{\left(2y+3\right)\left(5x^{2}-10xy+5y^{2}\right)}}{\frac{x\left(x-y\right)}{\left(x-y\right)^{2}}}$$

Cancel out x-y in both numerator and denominator.

$$\frac{\frac{5x\left(4y+6\right)}{\left(2y+3\right)\left(5x^{2}-10xy+5y^{2}\right)}}{\frac{x}{x-y}}$$

Divide \frac{5x\left(4y+6\right)}{\left(2y+3\right)\left(5x^{2}-10xy+5y^{2}\right)} by \frac{x}{x-y} by multiplying \frac{5x\left(4y+6\right)}{\left(2y+3\right)\left(5x^{2}-10xy+5y^{2}\right)} by the reciprocal of \frac{x}{x-y}.

$$\frac{5x\left(4y+6\right)\left(x-y\right)}{\left(2y+3\right)\left(5x^{2}-10xy+5y^{2}\right)x}$$

Cancel out x in both numerator and denominator.

$$\frac{5\left(4y+6\right)\left(x-y\right)}{\left(2y+3\right)\left(5x^{2}-10xy+5y^{2}\right)}$$

Factor the expressions that are not already factored.

$$\frac{2\times 5\left(2y+3\right)\left(x-y\right)}{5\left(2y+3\right)\left(x-y\right)^{2}}$$

Cancel out 5\left(2y+3\right)\left(x-y\right) in both numerator and denominator.

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