Its beautiful (Just observe it)

Algebra Level 3

{4x2+1x=5y2+1y=6z2+1zxyz=x+y+z\begin{cases} 4\dfrac{\sqrt{x^{2} + 1}}{x} = 5\dfrac{\sqrt{y^{2} + 1}}{y}=6\dfrac{\sqrt{z^{2} + 1}}{z} \\ xyz = x + y + z\end{cases}

If x=ab,y=cde,z=fgx = \dfrac{\sqrt{a}}{b} , y = c\dfrac{\sqrt{d}}{e} , z = f\sqrt{g} satisfy the above system of equations, then find (a+b++g) (a + b + \ldots + g ).

  • The expression is in simplest terms, meaning that all of the variables are positive integers, a,d,g a, d, g are square free and c,ec, e are coprime.
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