Question

Let $S$ be 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 $15\le x^2+y^2+z^2\le 150$. Then, the number of elements in the set $S$ is equal to

Answer 1

$x-2y+5z=0\cdots \left(i\right)$
$-2x+4y+z=0\cdots \left(ii\right)$
$-7x+14y+9z=0\cdots \left(iii\right)$

$2\times \left(i\right)+\left(ii\right)$
$\Rightarrow z=0$
$\Rightarrow x=2y$
$\Rightarrow 15\le 4y^2+y^2\le 150$
$\Rightarrow 3\le y^2\le 30$
$y\in [-\sqrt{30},-\sqrt{3}]\cup [\sqrt{3},\sqrt{30}]$
$y=\pm 2,\pm 3,\pm 4,\pm 5$
Number of elements in $S$ is 8.