How to Solve System of Linear Equations (Without Solver)?

I'm trying to solve for a system of 9 linear equations containing 12 variables. Thus, matrix algebra doing an inverse matrix won't work.

Does anyone have any suggestions as to how I might solve this system?

Many thanks for any ideas!

I find that, before I can make something like this an Excel problem, I have to make sure I can solve it as a math problem.

My first observation is that, with 12 unknowns and only 9 equations, a single solution doesn't likely exist. Usually you need at least as many equations as you have unknowns to get a unique solution.

The first thing I would probably do is double check to make sure I really have identified all of the equations/constraints for the problem. You need at least three more constraints/equations in order to get a unique solution, I'd start by looking for something that I've overlooked.

If I decided that I had not overlooked anything, I might start looking at the obvious substitutions and eliminations, and start "solving" the system by hand. In this way, I might be able to come up with a functional relationship that represents the solutions to the system.

I might also consider if there are any assumptions I can make about any of the variables. Sometimes the right assumption about a variable will significantly simplify the problem. The other suggestions really have very little to do with Excel, but this suggestion might allow you to use Excel. You can set up the spreadsheet to solve the system at whatever constraints you add to the problem. Once the spreadsheet is set up, it will be easy to change one (or more) of those constraints and get another solution. In this way, you might be able to get a picture of the solution space.

MrShorty: Thanks for your comments. I'll do as you suggested and take a closer look at the equations.

Thanks again!