Phasors of power supply Vs1 in one example
Well, at node 1 we have:

$$\frac{V_{S1} - V_1}{2 + j2} = \frac{V_1}{\frac{1}{2}||(-j2)} + \frac{V_1 - V_2}{j4}$$

which can be solved for \$V_{S1}\$.  Is this the equation you have?

If so, are you sure you've solved correctly for \$V_{S1}\$?

Also, is \$V_1 = 1 + j2\$?