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

Step-by-step solution

x - 2 y + 5 z = 0   . . . . . . . . . . . i - 2 x + 4 y + z = 0   . . . . . . . . . . i i - 7 x + 14 y + 9 z = 0   . . . . . . . . . . i i i Applying 2 × equation i + equation i i , we get z = 0 Putting z = 0 in equation i , we get x = 2 y Given 15 ≤ x 2 + y 2 + z 2 ≤ 150 ⇒ 15 ≤ 4 y 2 + y 2 ≤ 150 ⇒ 3 ≤ y 2 ≤ 30 ⇒ y = ± 2 ,   ± 3 ,   ± 4 ,   ± 5 ⇒ 8 solutions.