Evaluate
\frac{x-y}{x+y}
Expand
-\frac{y-x}{x+y}
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\frac{\frac{x+y}{\left(x+y\right)\left(x-y\right)}-\frac{x-y}{\left(x+y\right)\left(x-y\right)}}{\frac{2y}{x^{2}-2xy+y^{2}}}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-y and x+y is \left(x+y\right)\left(x-y\right). Multiply \frac{1}{x-y} times \frac{x+y}{x+y}. Multiply \frac{1}{x+y} times \frac{x-y}{x-y}.
\frac{\frac{x+y-\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}}{\frac{2y}{x^{2}-2xy+y^{2}}}
Since \frac{x+y}{\left(x+y\right)\left(x-y\right)} and \frac{x-y}{\left(x+y\right)\left(x-y\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{\frac{x+y-x+y}{\left(x+y\right)\left(x-y\right)}}{\frac{2y}{x^{2}-2xy+y^{2}}}
Do the multiplications in x+y-\left(x-y\right).
\frac{\frac{2y}{\left(x+y\right)\left(x-y\right)}}{\frac{2y}{x^{2}-2xy+y^{2}}}
Combine like terms in x+y-x+y.
\frac{2y\left(x^{2}-2xy+y^{2}\right)}{\left(x+y\right)\left(x-y\right)\times 2y}
Divide \frac{2y}{\left(x+y\right)\left(x-y\right)} by \frac{2y}{x^{2}-2xy+y^{2}} by multiplying \frac{2y}{\left(x+y\right)\left(x-y\right)} by the reciprocal of \frac{2y}{x^{2}-2xy+y^{2}}.
\frac{x^{2}-2xy+y^{2}}{\left(x+y\right)\left(x-y\right)}
Cancel out 2y in both numerator and denominator.
\frac{\left(x-y\right)^{2}}{\left(x+y\right)\left(x-y\right)}
Factor the expressions that are not already factored.
\frac{x-y}{x+y}
Cancel out x-y in both numerator and denominator.
\frac{\frac{x+y}{\left(x+y\right)\left(x-y\right)}-\frac{x-y}{\left(x+y\right)\left(x-y\right)}}{\frac{2y}{x^{2}-2xy+y^{2}}}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-y and x+y is \left(x+y\right)\left(x-y\right). Multiply \frac{1}{x-y} times \frac{x+y}{x+y}. Multiply \frac{1}{x+y} times \frac{x-y}{x-y}.
\frac{\frac{x+y-\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}}{\frac{2y}{x^{2}-2xy+y^{2}}}
Since \frac{x+y}{\left(x+y\right)\left(x-y\right)} and \frac{x-y}{\left(x+y\right)\left(x-y\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{\frac{x+y-x+y}{\left(x+y\right)\left(x-y\right)}}{\frac{2y}{x^{2}-2xy+y^{2}}}
Do the multiplications in x+y-\left(x-y\right).
\frac{\frac{2y}{\left(x+y\right)\left(x-y\right)}}{\frac{2y}{x^{2}-2xy+y^{2}}}
Combine like terms in x+y-x+y.
\frac{2y\left(x^{2}-2xy+y^{2}\right)}{\left(x+y\right)\left(x-y\right)\times 2y}
Divide \frac{2y}{\left(x+y\right)\left(x-y\right)} by \frac{2y}{x^{2}-2xy+y^{2}} by multiplying \frac{2y}{\left(x+y\right)\left(x-y\right)} by the reciprocal of \frac{2y}{x^{2}-2xy+y^{2}}.
\frac{x^{2}-2xy+y^{2}}{\left(x+y\right)\left(x-y\right)}
Cancel out 2y in both numerator and denominator.
\frac{\left(x-y\right)^{2}}{\left(x+y\right)\left(x-y\right)}
Factor the expressions that are not already factored.
\frac{x-y}{x+y}
Cancel out x-y 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}