Let S be the set of all integer solutions,

Question:

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 $15 \leq x^{2}+y^{2}+z^{2} \leq 150$. Then, the number of elements in the set $S$ is equal to__________.

Solution:

The given system of equations

$x-2 y+5 z=0$.....(1)

$-2 x+4 y+z=0$......(2)

$-7 x+14 y+9 z=0$.........(3)

From equation, $2 \times$ (1) $+$ (2) $\Rightarrow z=0$

Put $z=0$ in equation (1), we get $x=2 y$

$\because 15 \leq x^{2}+y^{2}+z^{2} \leq 150$

$\Rightarrow 15 \leq 4 y^{2}+y^{2} \leq 150$ $[\because x=2 y, z=0]$

$\Rightarrow 3 \leq y^{2} \leq 30$

$\Rightarrow y=\pm 2, \pm 3, \pm 4, \pm 5$

$\Rightarrow 8$ solutions.

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