Use Nodal Analysis to determine V₁ and V₂.

The original circuit is shown in black. The RED items were added.

I will sum the currents INTO the nodes.

First, note that on the left there are two resistors and a voltage source in series. Even though the two resistors are separated by the voltage source, we can still combine them in series to give a slightly simpler circuit without altering the values of the voltages at the essential nodes. (Can you develop a sensible argument to justify this?) This gives the circuit shown below.

Next, we must find V₅ so that we can get the current through the 5 W resistor into node A.

By KVL:

V_A = - V₅ + 640

so

V₅ = 640 - V_A

Now we need to write KCL at each essential node other than the reference node.

Note that the first term in the node A equation is really V₅/ 5; I just substituted using the expression we found for V₅ above.

Node A: (Eq. A)

Node B: (Eq. B)

Node C: (Eq. C)

Solving the three equations yields

V_A = 380 V

V_B = 269 V

V_C = 111 V

Finally,

V_A = V₁ = 380 V

V_B - V_A = V₂ = - 111 V

V_C = V₃ = 111 V