{ (((√(x/y)) +(√(y/z)) +(√(z/x)) = 3)),(((√(y/x)) +(√(z/y)) +(√(x/z)) = 3)),(((√(xyz)) = 1)) :}


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Question Number 183592 by greougoury555 last updated on 27/Dec/22

${\begin{cases} \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{z}} + \sqrt{\frac{z}{x}} = 3 \\ \sqrt{\frac{y}{x}} + \sqrt{\frac{z}{y}} + \sqrt{\frac{x}{z}} = 3 \\ \sqrt{x y z} = 1 \end{cases}$
	Answered by mr W last updated on 27/Dec/22

	$a = \sqrt{\frac{x}{y}}, b = \sqrt{\frac{y}{z}}, c = \sqrt{\frac{z}{x}}$ $a b c = 1$ $a + b + c = 3$ $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 3$ $a b + b c + c a = 3$ $a, b, c a r e r o o t s o f$ $t^{3} - 3 t^{2} + 3 t - 1 = 0$ ${(t - 1)}^{3} = 0$ $\Rightarrow a = b = c = t = 1$ $\Rightarrow x = y = z = 1 ✓$
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