I'm still having problems with supernodes. Correct me if my logic is wrong.
I took the KVL of the circuit which was \$-v_1 +2V_o + V_2 = 0\$;
Afterwards, I did the nodal analysis of both the incoming and outgoing currents from both sides, which I wrote out as
$$ 2 + \dfrac{v_2 - v_3}{8s} = \dfrac{v_1 -v_3}{2s} + \dfrac{v_1}{s} + \dfrac{v_2}{4s}; v_{13} = 13$$
next I made \$\dfrac{vs}{4s} = V_o\$. Is this the proper way to go about the problem?
I get \$16s - 39 = 12v_1 -v_2 +8sV_o\$ as the second equation.