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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.

enter image description here

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The first issue you have is that the values of your resistors are given in Siemens rather than Ohms, as shown by the units "S" instead of "\$\Omega\$". Siemens are the unit of conductance, the inverse of resistance.

So for each resistor in the circuit, you should be calculating

\$I = GV\$

instead of

\$I = V/R\$.

For example, the current through the 1 S resistor is \$(1\space\mathrm{S})\times{}v_1\$, not \$\frac{v_1}{1\space{}\Omega}\$. This will dramatically change your results.

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