Question

Let $(x,y,z)$ be points with integer coordinates satisfying the system of homogeneous equations:
$3x-y-z=0$
$-3x+z=0$
$-3x+2y+z=0$.
Then the number of such points which lie inside a sphere of radius $10$ centered at the origin is

Answer 1

Adding first two equations, we get $y=0$

and substituting $y=0$ in third equation, we get, $z=3x$

So any point which satisfies given system can be taken as, $(a,0,3a)$

Now for this point to lie inside inside a sphere of radius $10$ centered at origin.

$\Rightarrow a^2+0^2+(3a{)}^2<{10}^2$

$\Rightarrow a^2<10$

So, possible integral values of $a$ are $-3,-2,-1,0,1,2,3$

Hence, number of such points is $7$.