Why can't Mathematica "Solve" my equation?

Question

I am trying to get Mathematica to solve the following equality in terms of $w_i$

$\displaystyle \frac{\sum_{i = 1}^{n} x_i }{\sum_{i = 1}^{n} y_i} = \frac{\sum_{i = 1}^{n} ( (\frac{x_i}{y_i}) w_i ) }{n} $

(Background: https://math.stackexchange.com/questions/137332/relationship-between-ratios-and-averages-of-ratios)

But it's telling me "This system cannot be solved with the methods available to Solve."

Now, I'm guessing this is because I haven't given Mathematica enough information about my assumptions (specifically, that $w_i$ is expressible as a function of $x_i$ and $y_i$, or something similar)

What am I doing wrong here, and how would I go about using Mathematica to solve equalities like these?

Solve[Sum[Subscript[x, i], 
     {i, 1, n}]/
    Sum[Subscript[y, i], 
     {i, 1, n}] == 
   Sum[(Subscript[x, i]/
       Subscript[y, i])*
      Subscript[w, i], 
     {i, 1, n}]/n, 
  Subscript[w, i]]

PS.) I realize my title stinks... I blame myself for not even knowing enough about what my problem is to give this question a cogent title. If someone could Edit the question and give it a real title and/or suggest one, I'd be appreciative.

Answer 1

Mathematica can only handle this if you give an explicit value to n.

If I understand correctly, what you want is: for what values of $w_i$ in terms of $y_i$ is the equations going to be satisfied for any $x_i$.

You need to use SolveAlways, not Solve, for this.

Example:

eqn = Sum[Subscript[x, i], {i, 1, n}]/
    Sum[Subscript[y, i], {i, 1, n}] == 
   Sum[(Subscript[x, i]/Subscript[y, i])*Subscript[w, i], {i, 1, n}]/
    n;

num = 3; (* let's solve for n==3 *)

SolveAlways[eqn /. n -> num, Table[Subscript[x, i], {i, num}]]

The output is

$$ \left\{\left\{y_1\to 0,w_1\to 0\right\},\left\{y_1\to 0,y_2\to 0\right\},\left\{y_1\to 0,y_3\to 0\right\},\left\{y_1\to -y_2,y_3\to 0\right\},\left\{y_1\to -y_3,y_2\to 0\right\},\left\{y_2\to 0,w_2\to 0\right\},\left\{y_2\to 0,y_3\to 0\right\},\left\{y_2\to -y_3,y_1\to 0\right\},\left\{y_3\to 0,w_3\to 0\right\},\left\{w_1\to \frac{3 y_1}{y_1+y_2+y_3},w_2\to \frac{3 y_2}{y_1+y_2+y_3},w_3\to \frac{3 y_3}{y_1+y_2+y_3}\right\}\right\} $$

You'll see that only the last solution is different from 0.