Let  Sbe the set of all integer solutions x,y,z of the system of equations x-2y+5z=0,-2x+4y+z=0,-7x+14y+9z=0 such that 15≤x2+y2+z2≤150 then the number of  elements in the set S is equal to

The given system of equations x-2y+5z=0,-2x+4y+z=0,-7x+14y+9z=0 is x-2y+5z=0Observing the equations z=0,x=2y Given 15≤x2+y2+z2≤150 Substitute z=0,x=2yIt gives 15≤5y2≤1503≤y2≤30 Hence, y=±2,±3,±4,±5