Solve for $z$ in the following equation: $1-iz = -1 + iz$ (where $i^2 = -1$). Simplify your answer as much as possible.
To do that, I can add $iz$ to both sides and get $1 = -1 + 2iz$. Then I can add $1$ to both sides and get $2 = 2iz$. Next, I can divide both sides by $2i$ and get $\frac{1}{i} = z$. But this is not the simplest form of $z$, since it has an $i$ in the denominator. To simplify it, I can multiply the numerator and denominator by $i$ and get $z = \frac{i}{i^2}$. Since $i^2 = -1$, I can simplify further and get $z = \frac{i}{-1} = -i$. 