Although it might be a simple question, I'm struggling with something here:
Suppose you have 3 mass-points along the X axis, each one having masses $M_1, M_2, M_3$, at positions $x_1, x_2, x_3$, respectively ($x_1 < x_2 < x_3$).
Now I wish to find the position of the center of the mass, and to do that I must find the place ($x$ value) in which the Torque is null. Now comes the problem: I have four options here. The first is to consider that the center of mass is to the left of $x_1$. The second is to consider it is between $x_1$ and $x_2$. The third, between $x_2$ and $x_3$. The fourth, to the right of $x_3$. How am I supposed to know which one to pick? Also, even if I work out all possible options, at the end I will find that only one of them is not absurd, and that is surely the center of mass of the system, but how this is equivalent to the well-known mathematical definition for the center of mass (ponderating the masses and $x$ coordinates) ?
If I understand you correctly you would like to see how the coordinate of the center of mass arises from considering the torques.
For any point $x$ each mass $M_1,M_2,M_3$ creates a torque with respect to that point $x$ which sum up to a resulting torque $T_r$: $$(x-x_1) M_1 g + (x-x_2) M_2 g + (x-x_3) M_3 g = T_r$$
The center of mass is at $x_M$ such that $T_r = 0$: $$(x_M-x_1) M_1 g + (x_M-x_2) M_2 g + (x_M-x_3) M_3 g = 0 \Rightarrow x_M =\frac{x_1M_1+x_2M_2+x_3M_3}{M_1+M_2+M_3}$$ I hope that is what you wanted to see.