Calculate: ((1)/(x)+(1)/(y)) * (x-y)-(x+y) * ((1)/(x)+(1)/(y))

Expression: $\left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right) \times \left( x-y \right)-\left( x+y \right) \times \left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right)$

Factor out $\frac{ 1 }{ x }+\frac{ 1 }{ y }$ from the expression

$\left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right) \times \left( x-y-\left( x+y \right) \right)$

When there is a $-$ in front of an expression in parentheses, change the sign of each term of the expression and remove the parentheses

$\left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right) \times \left( x-y-x-y \right)$

Since two opposites add up to $0$, remove them from the expression

$\left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right) \times \left( -y-y \right)$

Collect like terms

$\left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right) \times \left( -2y \right)$

Multiplying a positive and a negative equals a negative: $\left( + \right) \times \left( - \right)=\left( - \right)$

$-\left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right) \times 2y$

Use the commutative property to reorder the terms

$-2y \times \left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right)$

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