Let $(x, y, z)$ be points with integer coordinates satisfying the system of homogeneous equations:

$3x-y-z=0$  ...(i)

$- 3x + z = 0$  ...(ii)

$-3x+2y+z = 0$  ...(iii)

Then, the number of such points for which $x^2 + y^2+z^2 ≤100$, is

7

Adding (i) and (ii) equations, we get $y = 0$. From equation (ii), we have $z = 3x$.

It is given that $x^2 + y^2+z^2 ≤100$

$∴x^2 ≤10$

$⇒-\sqrt{10} ≤ x ≤ \sqrt{10}⇒ x=± 3, ±2, ±1, 0$ [∵ x is an integer]

Hence, there are 7 points.