I'm kind of new to Mathematica and i'm trying to solve a Linear Algebra problem for my class of Linear Algebra of the course of Physics. Also, my first language is portuguese so forgive me for my not-so-perfect english.
The problem consists on a system of 7 linear equations and 12 variables (A, B, C, D, E, F, G, H, I, J, K, L), already solved by the Gaussian-Jacob Method, that is the following:
A = 33 - K - L
B = 1 + F - J
C = -15 - F + J + K + L
D = 15 + H - K
E = 16 - F - H + J + K
G = 34 - H - J - L
I = 18 - J - K
So, the system is possible but undetermined (with 5 degrees of freedom), being F, H, J, K and L the free variables.
Note: I'm using letters (A, B, ..., L) instead of X1, X2, ..., X12 because it's easier to write it like this here and because I don't know if the Xn notation is allowed on Mathematica.
Next, and this is the important part, are the 2 questions asked by my professor, and that's where I would like you to help me, please:
a) Find all the solutions of the system that satisfies the following condition: all variables, from A to L (or X1 to X12, depending on what you've called them) must be positive integers, i.e., A, B, ..., L ? IN ? natural numbers.
b) Find all the solutions of the system that satisfies, besides the condition of a), the condition that all the variables, from A to L (or X1 to X12, once again, depending on what you've called them) have to be different from one another and they all must be positive integers between 3 and 14 (inclusive, of course).
So, after some research, i found that a possible way to solve this type of system of linear inequalities is trough a method of elimination (analog to Gauss-Jordan's elimination method for systems of linear equations) named Fourier-Motzkin. But it's hardwork and i wanted to do it on the computer, but it's my first time on Mathematica.
I really need help solving this as the professor told us that the first one to solve would win a book, eheh.
I would really apreciate an answer, as my only goal as a future physicist is to unveil the secrets of the Cosmos to us all.
Thank you again.