Q:  Determine the voltage V_o by using Thevenin's theorem .

Sol:

            We will estimate V₀ by using Thevenin's theorem .

Step 1: Remove R_L

To remove R_L to estimate V_th .

Step 2:

                               V_x = 12 + V₁

                           At node 1

                                                V₁/2k + 2V_x/1k +V_x/2k= 0

                                                             V₁ + 4V_x + V_x =0

                        By putting the value of V₁  from above equation

                                                            V_x - 12 +4V_x +V_x =0

                                                                                6V_x =12

                                                                                  V_x = 12/6 = 2V = V_th

Step 3: Determining the value of R_th

In the circuit there is dependent source we have to short circuit the open circuit therefore we have here    

                                                            V_x = V₁ +12

                                                  V₁/2k + 2V_x/1k + V_x/2k + V_x/2k = 0

                                                             V_x - 12 +4V_x + V_x + V_x =0

                                                                        7V_x =12

                                                            I_sc = 12/7 x1/2 = 6/7 mA

                                                            R_th = V_th/ I_sc

                                                                  = 2/(6/7)m

                                                              R_th = 7/3 k = 2.33k

Step 4:   Determining unknown quantity.

After estimating V_th & R_th, re-inserting the load resistance R_L in the circuit in series along R_th and letting the V_th like a battery in series along these two resistances.

                                                        V₀  = 2 x 2k/2k + 2.33k =0. 92V