Hey guys, there's a complex number problem on a textbook here that I noticed something weird

a and b are real numbers
(a + 3i)(1 + 2i) = b + 5i
What's a + b?

At first I though this couldn't be solved because there were two variables and only one equation, but the solution was to simplify it to
a + (3 + 2a)i - 6 = b + 5i

And then make two different equations
a - 6 = b
(3 + 2a)i = 5i

Which leads to a being 1 and b being -5, therefore the answer is -4

I can see that this is allowed because a and b can't contain i so it can be treated as a different equation, but what are the implications of that?

Does all complex equations with two variables always have only one real solution? Is this something normal that I didn't pay attention?