3 Simultaneous Equations (Possibly Multiple or No Solutions)

Problem:

117x+390y+390z=a

144x+216y+384z=b

260x+520y+45z=c

x, y & z are all positives integers.

What are all the possible values for x, y & z?

Values of a, b & c are not important.

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Hi, I studied both Calculus & Statistics in high school but that's it. In my statistics class I learnt about solving these sorts of equations (I believe it actually belongs in Algebra) but those equations were simpler because a, b & c were known. Also, I learnt something about graphing simpler ones to find optimal solutions for situations (Linear Programming).

Now I believe that with the use of advanced understanding in what would be a 3D version of the 2D "Linear Programming" idea, that those possible values for x, y & z can be easily found. Or found that there is indeed, no solution.

This is my first post, so please, if I have made a mistake in posting in the wrong section - let me know. Thanks a lot in advance!