Solve the following systems of equations by the method of cross-multiplication:
$x (a - b + \frac{a b}{a - b}) = y (a - b - \frac{a b}{a + b})$
$x + y = 2 a^{2}$

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Solution

$x (a - b + \frac{a b}{a - b}) - y (a - b - \frac{a b}{a + b}) = 0$
& $x + y - {2 a}^{2} = 0$
$\frac{x}{({- 2 a}^{2}) [- (a - b - \frac{a b}{a + b})] - 0} = \frac{- y}{({- 2 a}^{2}) [(a - b + \frac{a b}{a - b})] - 0} = \frac{1}{1. (a - b + \frac{a b}{a - b}) + (a - b - \frac{a b}{a + b})}$
$\Rightarrow \frac{x}{\frac{{2 a}^{2} (a^{2} - b^{2} - a b)}{a + b}} = \frac{+ y}{{2 a}^{2} (\frac{a^{2} + b^{2} - a b}{a - b})} = \frac{1}{\frac{(a^{2} + b^{2} - a b)}{a - b} + \frac{(a^{2} - b^{2} - a b)}{a + b}}$
$\Rightarrow x = \frac{{2 a}^{2} (a^{2} - b^{2} - a b) (a - b)}{[(a + b) (a^{2} + b^{2} - a b) + (a - b) (a^{2} - b^{2} - a b)]}$ , $y = \frac{{2 a}^{2} (a^{2} + b^{2} - a b) (a + b)}{[(a + b) (a^{2} + b^{2} - a b) + (a - b) (a^{2} - b^{2} - a b)]}$
$= \frac{{2 a}^{2} (a - b) (a^{2} - b^{2} - a b)}{{2 a}^{3} - {2 a}^{2} b + {2 b}^{3}}$ $y = \frac{{2 a}^{2} (a + b) (a^{2} + b^{2} - a b)}{2 (a^{3} + b^{3} - a^{2} b)}$

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Similar questions

x (a - b + \frac{a b}{a - b}) = y (a + b - \frac{a b}{a + b}) x + y = 2 a^{2}

Solve each of the following systems of equations by the method of cross-multiplication:

$x (a - b + \frac{a b}{a - b}) = y (a + b - \frac{a b}{a + b})$

$x + y = 2 a^{2}$

\frac{x}{a} = \frac{y}{b} a x + b y = a^{2} + b^{2}

\frac{x}{a} + \frac{y}{b} = 2 a x - b y = a^{2} - b^{2}

\frac{x}{a} + \frac{y}{b} = a + b

\frac{x}{a^{2}} + \frac{y}{b^{2}} = 2

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