Equality of Complex Numbers

Two complex numbers are equal only when both their real parts and their imaginary parts match:

$a + bi = c + di \iff a = c \text{ and } b = d$

This turns one complex equation into two ordinary equations, which is how you solve for unknowns hidden inside a complex expression.

Example: if $x + 2i = 5 + yi$, matching real parts gives $x = 5$, and matching imaginary parts gives $y = 2$.