There are 3 complex numbers $a+bi$, $c+di$, and $e+fi$. If $b=1$, $e=-a-c$, and the sum of the numbers is $-i$, find $d+f$.
I notice that the problem involves complex numbers, which have a real part and an imaginary part. I also notice that the problem gives me some information about the real and imaginary parts of some of the numbers. I want to find $d+f$, which are the imaginary parts of $c+di$ and $e+fi$. To do that, I need to use the fact that the sum of the numbers is $-i$. I can write the sum of the numbers as $(a+bi)+(c+di)+(e+fi)=-i$. I can simplify this by grouping the real and imaginary parts together: $(a+c+e)+(b+d+f)i=-i$. Since the sum of two complex numbers is equal only if their real and imaginary parts are equal, I can set up a system of equations: $a+c+e=0$ and $b+d+f=-1$. I already know that $b=1$, so I can substitute that into the second equation and get $d+f=-2$. This is the answer I was looking for, so I don't need to solve for the other variables. 