Yes, the equation of the tetrahedron is fixed.

However, for a small change in "a" (holding "b" fixed) there must be a corresponding change in "c" so that the tetrahedron still passes through (1,2,3).

This is what that step is trying to find (using the chain rule).

If you solved for "c" in terms of "a" and "b", and took the partials, it would give the same answer.