Distribution of weight among massless strings hanging a rod of uniform density

Let the mass of a rod of uniform density be $W$ and its length be $l$. Several massless strings have been used to hang the rod at positions $x_1, x_2, …, x_n$ where $0 leq x_r leq l$. Find tensions on strings $F_1, F_2, … F_n$ with respect to $W, l, x_r$.

(I am not writing units to simplify stating the problem.)

Two Strings:
I tried this problem with 2 strings. In one case, I took $W=1, l=1, x_1=0, x_2=1$. I got $F_1=F_2=frac{1}{2}$ by solving the following equations:

$F_1+F_2 = W$ [tensions must support the weight of the rod]

$0 cdot F_1 + l cdot F_2 = frac{l}{2} cdot W$ [sum of moments of individual forces about any point (in this case, $x_1$) is equal to the moment of force of the resultant about that point *, here resultant is acting along the centre of gravity]

*I don’t even know if this statement I have made above is correct or not

I also tried the problem changing $x_2$ to $frac{3}{4}$. In this case, I got $F_1 = frac{1}{3}, F_2 = frac{2}{3}$.

Three Strings:
This is where my problem begins. For simplification, I took $W=1, l=1, x_1=0, x_2= frac{l}{2}, x_3 = l$. I made the following equations:

$Sigma F = W$ [sum of tensions must support weight]

$0 cdot F_1 + frac{l}{2} cdot F_2 + l cdot F_3 = frac{l}{2} cdot W$ [moment about $x_1$]

$-frac{l}{2} cdot F_1 + 0 cdot F_2 + frac{l}{2} cdot F_3 = 0 cdot W$ [moment about $x_2$]

But I can’t solve this system of equations. I tried changing the $x_2$ to $frac{3l}{4}$ but I am encountering the same problem. Since the system can’t be solved, it is likely that I am missing some important piece of information but I can’t find out what it is.

The Questions: My textbook does not have problems like these. I tried googling but none of the sites were helpful. I hope someone will be kind enough to answer the following questions:

• Is the way in which I am solving the two string problem correct?
• What am I doing wrong in the three string problem?
• What is the general solution to this problem?