Way to solve this circuit using Nodal Analysis

Your supernode should enclose both sources. Your KCL equation is then:

\$\dfrac{v_1}{2} + \dfrac{v_2}{4} + \dfrac{v_3}{3} = 0\$

Then, you need three more equations (you have four unknowns):

\$v_1 - v_2 = 10V \$

\$v_3 - v_2 = 5\Omega \cdot i \$

\$i = \dfrac{v_1}{2\Omega} \$

Solve to get:

\$v_1 = 3.043V\$

\$v_2 = -6.956V\$

\$v_3 = 0.6522V\$

\$i = 1.522A\$

As long as ohms law is satisfied where V=I*R nodal and mesh can be used. They are both ways of writing a system of equations for a circuit so you know the current in each loop or the voltage at every node.

While they both work it is usually easier to solve for the unknowns with one over the other depending on the situation. For instance if you have a circuit with a current source you already know the current through that loop where as if you have a voltage source you know the voltage at that node.

These techniques give us an elegant way of finding all voltages or currents associated with the circuit which means we know if the circuit we are using in practice will be able to handle the current or voltage applied to the load.