@Kitonum Like the other one i just asked, if you could turn this code nto "pseudo-code", you would make my day, seriously. Thank you one more time.

@Kitonum Hi there. I really aprecciated your answers but i can't present this to my professor like this as i have no idea what your code means. Is there a way you could, in a simple way, explain step-by-step the not so obvious parts of your code so a person with not so many knowledge in maple can "understand" what you did and how you got your results? That would be awesome. Thank you so much.

@Carl Love I think that when @Kitonum writes this...

if 1+F-J > 0 and 15+H-K > 0 and 18-J-K > 0 and 33-K-L > 0 and 34-H-J-L > 0 and -15-F+J+K+L > 0 and 16-F-H+J+K > 0 then n:=n+1: M[n]:=[F,H,J,K,L]: fi:

...he's ssuming that A, B, C, D, E, G and I must be >0 and not between 3 and 14 as the problem asks. Am I wrong?

@Carl Love The solutions given by @Kitonum don't tell me if the value of the variables A, B, C, D, E, G and I, are in between 3 and 14 as the other ones. So, is there a way to provide all the values for the 12 variables, and not only for the free ones (F, H, J, K and L) in the solution? Thank you!

@Kitonum I really aprecciate your effort to help me. The problem is that there are a few things that i think that I didn't explain well, so you're answers doesn't fully "satisfy" what my professor's asking. So, in a last effort, i've written the problem in the best way i can and, if you can, i would really aprecciate if you could help me, once and for all. Thank you so much.

------------------------------------\----------------------------------------
PROBLEM:

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.


Notes: 
(1) 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 X_n notation is allowed on Maple (i don't think so).
(2) I've noticed that Maple doesn't allow the letters D and I for variables, so a substitute for those might be P and Q, but the choice is yours and arbitrary.


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 ϵ 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, altought you're previous answers were good, i don't think they are the corrcet answer to the problem. One of the main reasons why that ocurred was, i believe, because I didn't explain correctly the problem, so I'm sorry for that But now that i've written the whole problem, could you please help me? Thank you one more time.

@Kitonum Dude that was awesome thanks. Another line from the problem ask us about all the solutions when all the variables (from A to L) have to be different from one another and they all must be positive integers between 3 and 14. How would you do it? I can't use Markiyan Hirnyk 6128 answer as i get the error:

Error, `DirectSearch` does not evaluate to a module

So i'm relying to use your "brute force method" to get all the solution. Could you help me, please? thank you a lot

@Kitonum How did you get to the bounds for all variables:  1<=J<=17 ,  1<=K<=17 ,  1<=L<=32 ,  1<=H<=32 ,  1<=F<=36. That's kind of what i wanted. Did you use Maple at all to do that or did you do it "on paper"? Thank you very much for all of your answers btw. I'm still trying the different methods but all i wanted was a simple answer like the one i'm asking about that you wrote.