((x+y)/x-2x/(x-y))×((y-x)/(x^2+y^2))

asked by guest
on Oct 25, 2024 at 6:19 am



You asked:

Evaluate the expression: \(\left(\frac{x + y}{x} - \frac{2 x}{x - y}\right) \cdot \frac{y - x}{{x}^{2} + {y}^{2}}\)

MathBot Answer:

Evaluated



\(\displaystyle \left(\frac{x + y}{x} - \frac{2 x}{x - y}\right) \cdot \frac{y - x}{{x}^{2} + {y}^{2}} = \frac{\left(- x + y\right) \left(- \frac{2 x}{x - y} + \frac{x + y}{x}\right)}{x^{2} + y^{2}} \)

Expanded

\[\left(\frac{x + y}{x} - \frac{2 x}{x - y}\right) \cdot \frac{y - x}{{x}^{2} + {y}^{2}} = \frac{2 x^{2}}{x^{3} - x^{2} y + y^{2} x - y^{3}} - \frac{2 x y}{x^{3} - x^{2} y + y^{2} x - y^{3}} - \frac{x}{x^{2} + y^{2}} + \frac{y^{2}}{x^{3} + y^{2} x}\]

Factored

\[\left(\frac{x + y}{x} - \frac{2 x}{x - y}\right) \cdot \frac{y - x}{{x}^{2} + {y}^{2}} = \frac{1}{x}\]

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