Question

rezolvați sistemul
$\sqrt{ \frac{x}{y} } + \sqrt{ \frac{y}{x} } = \frac{5}{2} \\ \frac{1}{x} + \frac{1}{y} = \frac{5}{16}$
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Answer 1

√(x/y) + √(y/x) = 5/2

Notam √(x/y) = t => √(y/x) = 1/t

=> t + 1/t = 5/2 |•2t

=> 2t²+2 = 5t => 2t²-5t+2= 0

Delta = 25 - 16 = 9

=> t1,2 = (5±3)/4

=> t1 = 1/2, t2 = 2

(1) t = 1/2 => √(x/y) = 1/2 => x/y = 1/4

1/x+1/y = 5/16 |•x

1+x/y = 5x/16

1+1/4 = 5x/16

16+4 = 5x => x = 4

4/y = 1/4 => y = 16

(2) t = 2 => √(x/y) = 2 => x/y = 4

1/x+1/y = 5/16 |•x

1+x/y = 5x/16

1+4 = 5x/16

5 = 5x/16

x = 16

=> 16/y = 4 => y = 4

=> (x,y) = {(4;16), (16;4)}