Let S be the set of all integer solutions, (x, y, z), of the system of equations

$x-2y+5z=0$
 $-2x+4y+z=0$
 $-7x+14y+9z=0$
such that $15\le x^2+y^2+z^2\le 150$  . Then, the number of elements in the set S is equal to _____________. 
 

$\begin{array}{l}\Delta =\left|\begin{array}{ccc}1& -2& 5\\ -2& 4& 1\\ -7& 14& 9\end{array}\right|=0\\ Letx=k\\ Putin\left(1\right)\&\left(2\right)\\ k-2y+5z=0\\ -2k+4y+z=0\\ z=0,y=\frac{k}{2}\\ \therefore x,y,zareinteger\\ \Rightarrow kiseveninteger\\ Nowx=k,y=\frac{k}{2},z=0 putincondition\\ 15\le k^2+{\left(\frac{k}{2}\right)}^2+0\le 150\\ 12\le k^2\le 120\\ \Rightarrow k=\pm 4,\pm 6,\pm 8,\pm 10\\ \Rightarrow NumberofelementinS=8.\end{array}$