System of linear inequalities (12 variables)

Questions
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1

Hi there.

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.

Miguel Jesus

2

You could use Solve in Mathematica. First avoid variables that begin with capital letters (use x1, x2,… or x[1],x[2],…). 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.