Let S be the set of all integer solutions x ,   y ,   z of the system of equations x - 2   y…

$x-2y+5z=0 ...........\left(i\right)$

$-2x+4y+z=0 ..........\left(ii\right)$

$-7x+14y+9z=0 ..........\left(iii\right)$

Applying $2\times$ equation $\left(i\right)+$equation $\left(ii\right)$, we get

$z=0$

Putting $z=0$ in equation $\left(i\right)$, we get

$x=2y$

Given $15\le x^2+y^2+z^2\le 150$

$\Rightarrow 15\le 4y^2+y^2\le 150$

$\Rightarrow 3\le y^2\le 30$

$\Rightarrow y=\pm 2, \pm 3, \pm 4, \pm 5$

$\Rightarrow 8$ solutions.