A bunch of expressions!

Algebra Level 3

( x x − y − x x + y ) ÷ ( y x − y − y x + y ) + ( x + y x − y + x − y x + y ) ÷ ( x + y x − y − x − y x + y ) = ? \left(\frac{x}{x-y}- \frac{x}{x+y} \right) \div \left(\frac{y}{x-y} - \frac{y}{x+y}\right) + \left(\frac{x+y}{x-y} + \frac{x-y}{x+y} \right) \div \left(\frac{x+y}{x-y} - \frac{x-y}{x+y} \right) = ?

Simplify the expression above.

4 x 2 + 2 y 2 2 x y \dfrac{4x^2 +2y^2}{2xy} 3 x 2 + y 2 2 x y \dfrac{3x^2 +y^2}{2xy} 2 x 2 + y 3 2 x y \dfrac{2x^2 +y^3}{2xy} x − y 2 x y \dfrac{x-y}{2xy} x + y 2 x y \dfrac{x +y}{2xy}

Munem Shahriar by Munem Shahriar , 18, Bangladesh

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1 solution

Chew-Seong Cheong
Sep 17, 2017

X = ( x x − y − x x + y ) ÷ ( y x − y − y x + y ) + ( x + y x − y + x − y x + y ) ÷ ( x + y x − y − x − y x + y ) = x x − y − x x + y y x − y − y x + y + x + y x − y + x − y x + y x + y x − y − x − y x + y = x ( 1 x − y − 1 x + y ) y ( 1 x − y − 1 x + y ) + ( x + y ) 2 + ( x − y ) 2 ( x − y ) ( x + y ) ( x + y ) 2 − ( x − y ) 2 ( x − y ) ( x + y ) = x y + 2 ( x 2 + y 2 ) 4 x y = 3 x 2 + y 2 2 x y \begin{aligned} X & = \left(\frac x{x-y} - \frac x{x+y} \right) \div \left(\frac y{x-y} - \frac y{x+y} \right) + \left(\frac {x+y}{x-y} + \frac {x-y}{x+y} \right) \div \left(\frac {x+y}{x-y} - \frac {x-y}{x+y} \right) \\ & = \frac {\dfrac x{x-y} - \dfrac x{x+y}}{\dfrac y{x-y} - \dfrac y{x+y}} + \frac {\dfrac {x+y}{x-y} + \dfrac {x-y}{x+y}} {\dfrac {x+y}{x-y} - \dfrac {x-y}{x+y}} \\ & = \frac {x\left(\dfrac 1{x-y} - \dfrac 1{x+y} \right)}{y\left(\dfrac 1{x-y} - \dfrac 1{x+y} \right)} + \frac {\dfrac {(x+y)^2+(x-y)^2}{(x-y)(x+y)}} {\dfrac {(x+y)^2-(x-y)^2}{(x-y)(x+y)}} \\ & = \frac xy + \frac {2(x^2+y^2)}{4xy} \\ & = \boxed{\dfrac {3x^2+y^2}{2xy}} \end{aligned}

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