Question

2m - 3n = 21 N = 3 - m

Answer 1

You have two relations for three unknowns, so there are infinitely many solutions.

From 2m − 3n = 21:
n = (2m − 21)/3 = (2/3)m − 7.

Also N = 3 − m.

So the general solution (with free parameter m = t) is:
m = t,
n = (2t − 21)/3,
N = 3 − t,
for any real t.

If you want integer solutions, m must be divisible by 3. Put m = 3k (k integer) to get:
m = 3k,
n = 2k − 7,
N = 3 − 3k.

Example: k = 4 → m = 12, n = 1, N = −9.