My boyfriend and I were also able to solve the system for one set. The symbols represent a numeric system in base seven. (values 0-6).
Warning: If you switch puzzles the mapping between symbols and values may change.
So you need to consider each puzzle individually and not close a puzzle and re-open it.
The first thing we did was look at the sums of 4 of the same symbol for all possible symbols. (not all symbols were present in my key).
If you can find a case where 4 of one symbol gives that same symbol then that symbol must be 0. For example if z + z + z + z = z then z = 0, similarly if z + z + z + v = v then z must be 0.
Once you know what symbol is 0 then you can start to work out 2 symbol equations and combine those to determine the other values. For example...
1. o + o + o + o = o so o = 0
2. o + o + c + v = T
3. o + o + c + B = o
4. o + o + c + T = x
5. o + o + c + x = B
We see that c + B = o = 0 so that's a good place to start. We can substitute for B using equation #5 and get:
c + B = c + c + x = 0 from there we can substitute in equation 4 and get:
c + c + x = c + c + c + T = 0, repeat with equation 2 and we get:
c + c + c + T = c + c + c + c + v = 0
Now since this is base 7 this really means that 4*c + v = 7 or some multiple of 7. The easiest solution here is to assume that c = 1 and v = 3 (although it turns out there are multiple possible solutions for any value of c).
If c = 1 and v = 3 then T = 4 (equation 2) so x = 5 (equation 4) and so on.
Hopefully that example makes sense. For the math nerds out there this is just a case of manually solving a system of linear equations and a better (and faster) solution would be to construct a sufficiently large matrix of equations and use
Gaussian elimination or
some other technique to solve for the symbol's values.
Ok so we now have a value for each of the symbols, unfortunately the puzzle isn't solved yet. Now you need to go back through all the possible modifiers and find a set that actually produce the correct result.
Edit:
There's probably a reasonable way to automate this. I'm trying to think of an elegant solution that would not require a user to solve their system, and then manually enter the numeric versions of their, key, solution and all possible modifiers (or at least the first few digits of each modifier) in order to brute-force a solution. I may work on that tomorrow.