How can I solve (nontrivially) this equation in nonnegative integers: $$ x - 2y + 3z - 4t = 0.$$
By inspection I found the set of solutions is: {(2,1,0,0),(4,0,0,1), (0,3,2,0), (1,0,1,1), (0,0,4,3), (0,1,2,1), (1,2,1,0),(6,1,0,1)}
1- Is my solution correct?
2- Also, I got a hint that this can be solved as a linear Diophantine equation but I only know how to do this for 2 variable and sometimes for 3 (where I express the third variable in terms of the other two)but not for 4 variables.
Any help will be appreciated.
Clearly there are infinitely many solutions; scaling any solution yields another solution.
As for explicit solutions;for any $y,t\geq0$ and any $z\geq0$ such that $3z\leq2y+4t$ you have $$x:=2y-3z+4t\geq0.$$