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

We have the three simultaneous linear equations

$$\begin{aligned} x-2y+5z&=0,\\ -2x+4y+z&=0,\\ -7x+14y+9z&=0. \end{aligned}$$

All variables are required to be integers and they must also satisfy the quadratic inequality

$$15\le x^{2}+y^{2}+z^{2}\le150.$$

To find every integer solution, we first study the linear system. We begin by observing a relationship among the equations. From the first equation we can isolate the expression $$x-2y$$:

$$x-2y=-5z.$$

Now we substitute this expression in the third equation. The left-hand side of the third equation may be rewritten as follows:

$$-7x+14y+9z = -7(x-2y)+9z.$$

Using $$x-2y=-5z$$ inside this, we get

$$-7(x-2y)+9z=-7(-5z)+9z=35z+9z=44z.$$

But the third equation states that this entire quantity equals zero, so we must have

$$44z=0\;\Longrightarrow\;z=0.$$

With $$z=0$$ forced, the first equation becomes

$$x-2y+5(0)=0\;\Longrightarrow\;x-2y=0\;\Longrightarrow\;x=2y.$$

Substituting $$x=2y$$ and $$z=0$$ into the second equation gives

$$-2(2y)+4y+0=0\;\Longrightarrow\;-4y+4y=0,$$

which is automatically satisfied, confirming that no additional restriction arises.

Therefore every integer solution of the linear system can be written in parameter form as

$$x=2t,\quad y=t,\quad z=0,$$

where $$t$$ is any integer.

Next, we impose the quadratic condition. For the parametrised solution we have

$$x^{2}+y^{2}+z^{2}=(2t)^{2}+t^{2}+0^{2}=4t^{2}+t^{2}=5t^{2}.$$

The requirement $$15\le x^{2}+y^{2}+z^{2}\le150$$ therefore becomes

$$15\le5t^{2}\le150.$$

Dividing every part of the inequality by $$5$$, we obtain

$$3\le t^{2}\le30.$$

To satisfy this, the square $$t^{2}$$ must be at least $$3$$ and at most $$30$$. Taking square-roots gives

$$\sqrt{3}\le|t|\le\sqrt{30}.$$

We know $$\sqrt{3}\approx1.732$$ and $$\sqrt{30}\approx5.477,$$ so the integer values of $$|t|$$ that lie in this range are

$$|t|=2,3,4,5.$$

Thus the permissible integer values of the parameter $$t$$ are

$$t=-5,-4,-3,-2,2,3,4,5.$$

Each value of $$t$$ gives exactly one ordered triple $$(x,y,z)=(2t,t,0)$$, and there are

$$8$$

such values of $$t$$ in total.

So, the answer is $$8$$.