Find the total power dissipated in the circuit below using the node voltage method.

The chosen node voltage variables are shown in brown; the voltage shown in blue was added to facilitate the solution.

I will sum the currents out of the nodes.

There is a slight difficulty with node A due to the voltage source in series with the 12 ohm resistor. I have added the variable V₁₂ to help with determining the current out of node A through the 12 ohm resistor. Not that the polarity of V₁₂ was chosen so that the current leaving the node enters the positive reference.

We know that the total voltage from node A to the reference node (which is just V_A) must equal the summ of the voltages across the 12 ohm resistor and the voltage source, thus

V_A = V₁₂ - 40

so

V₁₂ = V_A + 40

Now we can write KCL at node A

Multiply through by 300

25 V_A + 1000 + 12 V_A + 15 V_A - 15 V_B + 1500 = 0
52 V_A - 15 V_B + 0 V_C = - 2500 (Equation 1)

Multiply through by 40

2 V_B - 2 V_A + V_B - V_C - 300 - 200 = 0
- 2 V_A + 3 V_B - V_C = 500 (Equation 2)

Multiply through by 40

V_C - V_B + V_C + 300 = 0
0 V_A - V_B + 2 V_C = - 300 (Equation 3)

Solving the three equations gives

V_A = - 10
V_B = 132
V_C = - 84

First the resistors.

12 Ohm

P₁₂ = (V₁₂)² / 12 = 30² / 12
P₁₂ = 75 W

20 Ohm

P₂₀ = (V_A - V_B)² / 20 = (-142)² / 20
P₂₀ = 1008.2 W

25 Ohm

P₂₅ = V_A² / 25 = (-10)² / 25
P₂₅ = 4 W

40 Ohm (right side)

P_40A = (V_B - V_C)² / 40 = 216² / 40
P_40A = 1166.4 W

40 Ohm (bottom)

P_40B = V_C² / 40 = (-84)² / 40
P_40B = 176.4 W

The total power dissipated by the resistors is the sum of the above, or

P_R = 2430 W

Now we need to determine if any of the sources are actually absorbing power.

The power generated by the 40 Volt source is 40 (V₁₂ / 12) = 40 (-10 + 40) / 12
Since this is obviously positive, the 40 volt source is NOT absorbing power.

The power generated by the 5 amp source is (V_B - V_A) (5) = (132 + 10) (5)
This is also positive, thus the 5 amp source is NOT absorbing power.

The power generated by the 7.5 amp source is (V_B - V_C) (7.7) = (132 + 84) (7.5)
This again is positive, so the 7.5 amp source is NOT absorbing power.

The total power absorbed in the circuit is thus equal to the total power absorbed by the resistors which we found above to be

P_R = 2430 W