Systems of Equations not properly explained about quantumkatas HOT 1 CLOSED

The solution to this system of equations is not unique, you can multiply the $\alpha$ and $\beta$ given as the solutions by any complex number with absolute value 1 and $\gamma$ and $\delta$ - by its adjoint and you'll get a valid solution as well. In practice, these solutions differ from the given one only by the global phases of the qubit states, which is not observable, so they are ignored.

To solve the system, you can fix $\alpha$ to be a non-negative real number, and, taking into account the normalization requirement for both qubit states ($|\alpha|^2 + |\beta|^2 = 1$ and $|\gamma|^2 + |\delta|^2 = 1$), find out that the absolute value of each of those numbers is $\frac1{\sqrt2}$; after this, you only need to adjust the relative phases of each number.

from quantumkatas.