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

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. ⇒a² + 0² + (-3a)² < 10² ⇒10a² < 100 ⇒a² < 10. So, possible integral values of a are -3, -2, -1, 0, 1, 2, 3. Hence, number of such points is 7.