Question

Let $(x, y, z)$ be points with integer coordinates satisfying the system of homogeneous equations:
$3 x-y-z=0 $
$-3 x+z=0$
$-3 x+2 y+z=0$
Then the number of such points for which $x^{2}+y^{2}+z^{2} \leq 100$ is

Answer 1

$3 x-y-z=0 $
$-3 x+z=0 $
$\Rightarrow y=0$
and $ z=3 x$
$\Rightarrow x^{2}+y^{2}+z^{2}=x^{2}+z^{2}$
$=x^{2}+9 x^{2}=10 x^{2} \leq 100 $
$\Rightarrow x^{2} \leq 10 $
$\Rightarrow x=0, \pm 1, \pm 2, \pm 3$
There are such seven points.