MHB Can You Solve This Week's Challenging System of Equations?

[anemone]

Here is this week's POTW:

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Solve the system of equation

$3x+7y+14z=252\\xyz-u^2=2016$

for non-negative real numbers.

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[anemone]

No one answered last week's POTW. (Sadface) However, you can find the suggested model solution as follows:

Spoiler

From the conditions of the problem we obtain

$\begin{align*} 252=3x+7y+14z& \ge 3\sqrt[3]{3x(7y)(14z)}\\&=3\sqrt[3]{3(7)(14)(2016+u^2)}\\& \ge 3\sqrt[3]{3(7)(14)(2016)}\\&=3\sqrt[3]{2^6(3^3)(7^3)}\\&=2^2(3^2)(7)\\&=252\end{align*}$

Equality is attainable when $3x=7y=14z$ and $u=0$. From the first equation of the system, we obtain

$3x=7y=14z=\dfrac{252}{3}=84$. This implies $x=28,\,y=12$ and $z=6$.