Question

5(1/x2+1/y2+1/z^2)=4(1/xy+1/yz+1/zx)

Answer 1

Algebra,

We have,

$5 \times (\frac{1}{ {x}^{2} } + \frac{1}{ {y}^{2} } + \frac{1}{ {z}^{2} } ) = 4 \times ( \frac{1}{xy} + \frac{1}{yz} + \frac{1}{xz} )$

1/x²+ 1/y²+ 1/z²= 4/5(1/xy + 1/yz + 1/xz)

(1/x + 1/y + 1/z)²-2(1/xy + 1/yz + 1/xz) = 4/5(1/xy + 1/yz + 1/xz)

(1/x + 1/y + 1/z)²=4/5(1/xy + 1/yz + 1/xz) + 2(1/xy + 1/yz + 1/xz)

(1/x + 1/y + 1/z)²=14/5(1/xy + 1/yz + 1/xz)

so the ansewr is,
$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \sqrt{\frac{14}{5} ( \frac{1}{xy} + \frac{1}{yz} + \frac{1}{xz})}$

or, 1/x + 1/y + 1/z = 1/xyz√{14/5(xy + yz + xz)}

I think it's the correct, infact sure it's the answer.

That's it
Hope it helped (●´ϖ`●)