<s>[INST] How to solve system of equation, $\sqrt{x-1}+\sqrt{y-1}=4\sqrt 3$, $\sqrt{y-4}+\sqrt{z-4}=4\sqrt3$ and $ \sqrt{x-9}+\sqrt{z-9}=4\sqrt3$ . $$\begin{cases}\sqrt{x-1}+\sqrt{y-1}=4\sqrt 3\\\sqrt{y-4}+\sqrt{z-4}=4\sqrt3\\\sqrt{x-9}+\sqrt{z-9}=4\sqrt3\end{cases}$$ I tried somthing,like go to the power of two , and change of variables... but it became more complicated . Is there an idea to solve this system of equation ?
  Thanks in advance 
 [/INST] Solving $1$:-
$\sqrt{x-1}+\sqrt{y-1}=4\sqrt3\\$
$\text{Squaring}$ 
$\begin{align}x-1+y-1+2\sqrt{(x-1)(y-1)}&=48\\to\\50-x-y&=2\sqrt{(x-1)(y-1)}\\\end{align}$
$\text{Squaring}$ 
$\begin{align}x^2  + y^2+ 2 x y- 100 x  - 100 y + 2500&=4 x y - 4 x - 4 y + 4\\to\\x^2  + y^2 + 2496 &=   96 x+ 96 y +2 x y \end{align}$
Continue for the other equations, then solve.
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