Question 2.11: Write an independent set of equations for the node voltages ...

First, we ignore the fact that the voltage source is a dependent source and write equations just as we would for a circuit with independent sources. We cannot write a current equation at either node 1 or node 2, because of the voltage source connected between them. However, we can write a KVL equation:

−v_1 + 0.5v_x + v_2 = 0        (2.46)

Then, we use KCL to write current equations. For a supernode enclosing the
controlled voltage source,

For node 3,

\frac{v_3}{R_4}+\frac{v_3-v_2}{R_3} + \frac{v_3-v_1}{R_1}=0        (2.47)

For the reference node,

\frac{v_1}{R_2} +\frac{v_3}{R_4}= i_s       (2.48)

Of course, these current equations are dependent because we have used all four nodes in writing them. We must use Equation 2.46 and two of the KCL equations to form an independent set. However, Equation 2.46 contains the controlling variable v_x, which must be eliminated before we have equations in terms of the node voltages.
Thus, our next step is to write an expression for the controlling variable v_x in terms of the node voltages. Notice that v_1, v_x, and v_3 form a closed loop. Traveling clockwise and summing voltages, we have

−v_1 − v_x + v_3 = 0
Solving for v_x, we obtain
v_x = v_3 − v_1
Now if we substitute into Equation 2.46, we get
v_1 = 0.5(v_3 − v_1) + v_2     (2.49)
Equation 2.49 along with any two of the KCL equations forms an independent set that can be solved for the node voltages.