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.
You could use Solve in Mathematica. First avoid variables that begin with capital letters (use x1, x2,... or x,x,...). Provide both equations and inequalities to Solve. For integer solutions, provide a domain specification (look up Solve in the Documentation Center to see what this means; I'm not going to do the full homework assignment).
As for part (b), I'm going to leave that alone.