Evaluate
\frac{x}{x-y}
Expand
-\frac{x}{y-x}
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\left(\frac{x}{x-y}+\frac{2xy}{\left(x-y\right)^{2}}\right)\left(\frac{2x}{x+y}-1\right)
Factor x^{2}-2xy+y^{2}.
\left(\frac{x\left(x-y\right)^{2}}{\left(x-y\right)\left(x-y\right)^{2}}+\frac{2xy\left(x-y\right)}{\left(x-y\right)\left(x-y\right)^{2}}\right)\left(\frac{2x}{x+y}-1\right)
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{x}{x-y} times \frac{\left(x-y\right)^{2}}{\left(x-y\right)^{2}}. Multiply \frac{2xy}{\left(x-y\right)^{2}} times \frac{x-y}{x-y}.
\frac{x\left(x-y\right)^{2}+2xy\left(x-y\right)}{\left(x-y\right)\left(x-y\right)^{2}}\left(\frac{2x}{x+y}-1\right)
Since \frac{x\left(x-y\right)^{2}}{\left(x-y\right)\left(x-y\right)^{2}} and \frac{2xy\left(x-y\right)}{\left(x-y\right)\left(x-y\right)^{2}} have the same denominator, add them by adding their numerators.
\frac{x^{3}-2x^{2}y+xy^{2}+2x^{2}y-2xy^{2}}{\left(x-y\right)\left(x-y\right)^{2}}\left(\frac{2x}{x+y}-1\right)
Do the multiplications in x\left(x-y\right)^{2}+2xy\left(x-y\right).
\frac{x^{3}-xy^{2}}{\left(x-y\right)\left(x-y\right)^{2}}\left(\frac{2x}{x+y}-1\right)
Combine like terms in x^{3}-2x^{2}y+xy^{2}+2x^{2}y-2xy^{2}.
\frac{x\left(x+y\right)\left(x-y\right)}{\left(x-y\right)\left(x-y\right)^{2}}\left(\frac{2x}{x+y}-1\right)
Factor the expressions that are not already factored in \frac{x^{3}-xy^{2}}{\left(x-y\right)\left(x-y\right)^{2}}.
\frac{x\left(x+y\right)}{\left(x-y\right)^{2}}\left(\frac{2x}{x+y}-1\right)
Cancel out x-y in both numerator and denominator.
\frac{x\left(x+y\right)}{\left(x-y\right)^{2}}\left(\frac{2x}{x+y}-\frac{x+y}{x+y}\right)
To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x+y}{x+y}.
\frac{x\left(x+y\right)}{\left(x-y\right)^{2}}\times \frac{2x-\left(x+y\right)}{x+y}
Since \frac{2x}{x+y} and \frac{x+y}{x+y} have the same denominator, subtract them by subtracting their numerators.
\frac{x\left(x+y\right)}{\left(x-y\right)^{2}}\times \frac{2x-x-y}{x+y}
Do the multiplications in 2x-\left(x+y\right).
\frac{x\left(x+y\right)}{\left(x-y\right)^{2}}\times \frac{x-y}{x+y}
Combine like terms in 2x-x-y.
\frac{x\left(x+y\right)\left(x-y\right)}{\left(x-y\right)^{2}\left(x+y\right)}
Multiply \frac{x\left(x+y\right)}{\left(x-y\right)^{2}} times \frac{x-y}{x+y} by multiplying numerator times numerator and denominator times denominator.
\frac{x}{x-y}
Cancel out \left(x+y\right)\left(x-y\right) in both numerator and denominator.
\left(\frac{x}{x-y}+\frac{2xy}{\left(x-y\right)^{2}}\right)\left(\frac{2x}{x+y}-1\right)
Factor x^{2}-2xy+y^{2}.
\left(\frac{x\left(x-y\right)^{2}}{\left(x-y\right)\left(x-y\right)^{2}}+\frac{2xy\left(x-y\right)}{\left(x-y\right)\left(x-y\right)^{2}}\right)\left(\frac{2x}{x+y}-1\right)
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{x}{x-y} times \frac{\left(x-y\right)^{2}}{\left(x-y\right)^{2}}. Multiply \frac{2xy}{\left(x-y\right)^{2}} times \frac{x-y}{x-y}.
\frac{x\left(x-y\right)^{2}+2xy\left(x-y\right)}{\left(x-y\right)\left(x-y\right)^{2}}\left(\frac{2x}{x+y}-1\right)
Since \frac{x\left(x-y\right)^{2}}{\left(x-y\right)\left(x-y\right)^{2}} and \frac{2xy\left(x-y\right)}{\left(x-y\right)\left(x-y\right)^{2}} have the same denominator, add them by adding their numerators.
\frac{x^{3}-2x^{2}y+xy^{2}+2x^{2}y-2xy^{2}}{\left(x-y\right)\left(x-y\right)^{2}}\left(\frac{2x}{x+y}-1\right)
Do the multiplications in x\left(x-y\right)^{2}+2xy\left(x-y\right).
\frac{x^{3}-xy^{2}}{\left(x-y\right)\left(x-y\right)^{2}}\left(\frac{2x}{x+y}-1\right)
Combine like terms in x^{3}-2x^{2}y+xy^{2}+2x^{2}y-2xy^{2}.
\frac{x\left(x+y\right)\left(x-y\right)}{\left(x-y\right)\left(x-y\right)^{2}}\left(\frac{2x}{x+y}-1\right)
Factor the expressions that are not already factored in \frac{x^{3}-xy^{2}}{\left(x-y\right)\left(x-y\right)^{2}}.
\frac{x\left(x+y\right)}{\left(x-y\right)^{2}}\left(\frac{2x}{x+y}-1\right)
Cancel out x-y in both numerator and denominator.
\frac{x\left(x+y\right)}{\left(x-y\right)^{2}}\left(\frac{2x}{x+y}-\frac{x+y}{x+y}\right)
To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x+y}{x+y}.
\frac{x\left(x+y\right)}{\left(x-y\right)^{2}}\times \frac{2x-\left(x+y\right)}{x+y}
Since \frac{2x}{x+y} and \frac{x+y}{x+y} have the same denominator, subtract them by subtracting their numerators.
\frac{x\left(x+y\right)}{\left(x-y\right)^{2}}\times \frac{2x-x-y}{x+y}
Do the multiplications in 2x-\left(x+y\right).
\frac{x\left(x+y\right)}{\left(x-y\right)^{2}}\times \frac{x-y}{x+y}
Combine like terms in 2x-x-y.
\frac{x\left(x+y\right)\left(x-y\right)}{\left(x-y\right)^{2}\left(x+y\right)}
Multiply \frac{x\left(x+y\right)}{\left(x-y\right)^{2}} times \frac{x-y}{x+y} by multiplying numerator times numerator and denominator times denominator.
\frac{x}{x-y}
Cancel out \left(x+y\right)\left(x-y\right) in both numerator and denominator.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}