Question:
Let $S$ be the set of all integer solutions, $(x, y, z)$, of the system of equations
$x-2 y+5 z=0$
$-2 x+4 y+z=0$
$-7 x+14 y+9 z=0$
such that $15 \leq x^{2}+y^{2}+z^{2} \leq 150$. Then, the number of elements in the set $S$ is equal to__________.
Solution:
The given system of equations
$x-2 y+5 z=0$.....(1)
$-2 x+4 y+z=0$......(2)
$-7 x+14 y+9 z=0$.........(3)
From equation, $2 \times$ (1) $+$ (2) $\Rightarrow z=0$
Put $z=0$ in equation (1), we get $x=2 y$
$\because 15 \leq x^{2}+y^{2}+z^{2} \leq 150$
$\Rightarrow 15 \leq 4 y^{2}+y^{2} \leq 150$ $[\because x=2 y, z=0]$
$\Rightarrow 3 \leq y^{2} \leq 30$
$\Rightarrow y=\pm 2, \pm 3, \pm 4, \pm 5$
$\Rightarrow 8$ solutions.