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Let's make it easier, add both of them you get  \frac{1}{x} +  \frac{1}{y} = 1 and subtract one from other,  \frac{1}{x} - \frac{1}{y} = 3. Now you get x=1/2 and y=-1.
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41/x-47/y=35  (eq1)
           47/x-41/y=53  (eq2)
  Now add eq1 &eq2, we get
     41(1/x-1/y)+47(1/x-1/y) = 83
 =>1/x=1+1/y (eq3)
Therefore by eq3 in eq1 we get x=1/2 & y=1    

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