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 
 A: So here are the 3 equations:
$$\begin{cases}\sqrt{x-1}+\sqrt{y-1}=4\sqrt 3\\\sqrt{y-4}+\sqrt{z-4}=4\sqrt3\\\sqrt{x-9}+\sqrt{z-9}=4\sqrt3\end{cases}$$
As suggested by transcenmental,
$\sqrt{x-1}-\sqrt{y-1}=4\sqrt 3 - 2 \sqrt{y-1}$ and multiplying with the first eq. gives 
$$
x - y = 4\sqrt 3 (4\sqrt 3 - 2 \sqrt{y-1})
$$
For the second eq., use
$$-\sqrt{y-4}+\sqrt{z-4}=4\sqrt 3 - 2 \sqrt{y-4}$$ and multiplying with the second eq. 
$$
z - y = 4\sqrt 3 (4\sqrt 3 - 2 \sqrt{y-4})
$$
Plugging into the last one gives an eq. in y:
$$
\sqrt{y + 4\sqrt 3 (4\sqrt 3 - 2 \sqrt{y-1})-9}+\sqrt{ y + 4\sqrt 3 (4\sqrt 3 - 2 \sqrt{y-4})-9}=4\sqrt3
$$
This is pretty akward, but $y = 28/3$ is a solution (by computer). From here the others follow, namely 
$$
x = 52/3$$ 
and 
$$
z = 76/3$$ 
EDIT:
with a little bit of hindsight and a little bit of psychology, you could argue 
as follows (with a twinkling of an eye):
suppose the person asking the question prefers a reasonably nicely looking solution (psychology 1). Then  all variables  should either be multiples of 3 or of $1/3$ to get rid of the $\sqrt 3$ on the RHS. Let's try $1/3$ (hindsight 1). So let $x = x' / 3$ etc. Now assume further that also the numerator of the variables should be nice, e.g. no roots etc. (psychology 2). Then we should have that $x'-3\cdot 1$ and  $x'-3\cdot 9$ are "nice" squares (likewise with the other variables). If we want it even nicer, they should be squares of integers (hindsight 2).
So $x' = 3 + n^2$ and $x' = 27 + m^2$. Now start playing. "Nice" integers n and m will be reasonably small (psychology 3). $x' = 52$ does it nicely, with $n=7$ and $m=5$. A small number of trials, to match all three variables, will then give the solution....
A: Hint:
Eliminate $y$ and $z$,
$$y=(4\sqrt3-\sqrt{x-1})^2+1,\\z=(4\sqrt3-\sqrt{x-9})^2+9$$
and 
$$\sqrt{44+x-8\sqrt3\sqrt{x-1}}+\sqrt{44+x-8\sqrt3\sqrt{x-9}}=4\sqrt3.$$
By plotting, you can see that this equation has two real solutions. It is possible, by successive squarings and regroupings, to turn it to a polynomial. But this will be tedious.
A: 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.
