What were the problems, other than time and memory?
If you can make all the parameters undefined symbols, integers, or ratios of integers, try Solve or Reduce.
If the parameters in the equations don't quite allow a solution, maybe you could turn the equations into error expressions
add them up, and use NMinimize or FindMinimum.
A notebook showing your inputs and any warning or error messages you get would help the rest of us help.