how to find the solution to this system of equations Given the system of equations:
$
xy+xz=54+x^2  \\ 
yx+yz=64+y^2 \\ 
zx+zy=70+z^2 $
Need to find all of the solutions of $ x,y$ and $z$.
Tried to sum up all three equations but got stuck with nothing to factorize. 

First assume $x,y,z\ne 0$.
From equation (1), $y+z=\dfrac {54}{x}+x\Rightarrow y+z-x=\dfrac {54}{x}$     ..........(A1)
From equation (2), $z+x=\dfrac {64}{y}+y\Rightarrow z+x-y=\dfrac {64}{y}$     ..........(A2)
From equation (3), $x+y=\dfrac {70}{z}+z\Rightarrow x+y-z=\dfrac {70}{z}$............(A3)
Now, adding (A1) & (A2), $2xyz=54y+64x$......(B1)
Adding (A2) & (A3), $2xyz=64z+70y$........(B2) 
Adding (A3) & (A1), $2xyz=70x+54z$...........(B3)
Equating (B1) & (B2), $64x-16y-64z=0$
Equating (B2) & (B3), $70x-70y-16z=0$
Equating (B3) & (B1), $16x-54y-54z=0$.
Now solve these equations.
