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

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

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

$- 7x + 14y + 9z = 0 . . . . . . . . . . \left(iii\right)</p><p>Applying$2 \times$equation \left(i\right) +$equation \left(ii\right)$, we get</p><p>$z = 0$</p><p>Putting$z = 0$in equation \left(i\right)$, we get

$x = 2y$

Given $15 \le {x}^{2} + {y}^{2} + {z}^{2} \le 150$

$⇒ 15 \le 4{y}^{2} + {y}^{2} \le 150$

$⇒ 3 \le {y}^{2} \le 30$

$⇒ y = \pm 2, \pm 3, \pm 4, \pm 5$

$⇒ 8$ solutions.