
I will choose node d as the reference node and assign node voltages Va, Vb, and Vc to nodes a, b, and c respectively. Thus the equations are:
KCL at node A:
First, determine V2a so that we can get an expression for the current into node A through the 2 W resistor.
VB - VA = - 110 + V2a
V2a = VB - VA + 110
Now write KCL into node A
Simplify and collect terms.
8 VB - 8 VA + 880 + 2 VC - 2 VA - VA = 0
- 11 VA + 8 VB + 2 VC = - 880 (Eq. A)
KCL at node B:
The current into node B through the upper 110 V source is simply the negative of the current through the 2 W resistor found above, so I'll skip the gory details. The current into node B through the lower 110 V source requires that we find V2b, however.
- VB = V2a - 110
V2a = 110 - VB
Now KCL into node B
Simplify and collect terms
3 VA - 3 VB - 330 + 330 - 3 VB + 2 VC - 2 VB = 0
3 VA - 8 VB + 2 VC = 0 (Eq. B)
KCL at node C:
Simplify and collect terms.
3 VA - 3 VC + 8 VB - 8 VC - VC = 0
3 VA + 8 VB - 12 VC = 0 (Eq. C)
Solving Equations A, B, and C yields:
VA = 157.1 V
VB = 82.5 V
VC = 94.29 V