Q:       Determine the value of voltage Va.

Solution:     We have to calculate the voltage Va. The circuit may be redrawn like

Va may also be determined by super node technique but it will take a heavy calculation let's see how it becomes very simple with path analysis.

                        Here from the circuit

                                                I₄ =1A

                        By Applying KVL for mesh 1

                                    1(I₁ - I₂) +3(I₁ -3I₃)-5 =0

                                           I₁ - I₂ +3I₁ -3I₃-5 = 0

                                              4I₁ - 3I₃ - I₂ = 5 ........1

                        By applying KVL for mesh 2

                                                2I₂ +3I₂ -2Va+I₂-I₁ = 0

                                                        6I₂ -I₁ = 2Va........2.

                                                                            Va = 3I₂

                        By putting this value in (1)

                                                            6I₂ -I₁ = 3I₂

                                                                    I₁ = 0

Applying KVL for mesh 3

                        2Va + 5(I₃ - I₄) +3(I₃ -I₁) =0

                        Put the values of I₁,I ₄and Va in the above equation

                                    6I₂ + 3I₃ -0 +5I₃ - 5 = 0

                                                                  8I₃ + 6I₂ = 5

                        By solving out equation for mesh 1 & mesh 3

                                                 - 18I₃ - 6I₂  = 30

                                        8I₃ + 6I₂  = 5

                                               -10I₃          = 35

                                                           I₃ = - 3.5A

                        By Putting values of  I₃ & I₁ in equation of  mesh 1 that is in (1) we obtain

                                                            10.5- I₂ =5

                                                     I₂= 5.5 A

As from above calculation                  V_a =3I₂

                        Hence                       V_a =3 * 5.5

                                                             V_a = 16.5 Volts