<P> (V 1 − V B R 1 + V 2 − V B R 2 + V 2 R 3 = 0 V 1 = V 2 + V A (\ displaystyle (\ begin (cases) (\ frac (V_ (1) - V_ (\ text (B))) (R_ (1))) + (\ frac (V_ (2) - V_ (\ text (B))) (R_ (2))) + (\ frac (V_ (2)) (R_ (3))) = 0 \ \ V_ (1) = V_ (2) + V_ (\ text (A)) \ \ \ end (cases))) </P> <P> By substituting V to the first equation and solving in respect to V, we get: </P> <P> V 2 = (R 1 + R 2) R 3 V B − R 2 R 3 V A (R 1 + R 2) R 3 + R 1 R 2 (\ displaystyle V_ (2) = (\ frac ((R_ (1) + R_ (2)) R_ (3) V_ (\ text (B)) - R_ (2) R_ (3) V_ (\ text (A))) ((R_ (1) + R_ (2)) R_ (3) + R_ (1) R_ (2)))) </P> <P> In general, for a circuit with N (\ displaystyle N) nodes, the node - voltage equations obtained by nodal analysis can be written in a matrix form as derived in the following . For any node k (\ displaystyle k), KCL states ∑ j ≠ k G j k (v k − v j) = 0 (\ displaystyle \ sum _ (j \ neq k) G_ (jk) (v_ (k) - v_ (j)) = 0) where G k j = G j k (\ displaystyle G_ (kj) = G_ (jk)) is the negative of the sum of the conductances between nodes k (\ displaystyle k) and j (\ displaystyle j), and v k (\ displaystyle v_ (k)) is the voltage of node k (\ displaystyle k). This implies 0 = ∑ j ≠ k G j k (v k − v j) = ∑ j ≠ k G j k v k − ∑ j ≠ k G j k v j = G k k v k − ∑ j ≠ k G j k v j (\ displaystyle 0 = \ sum _ (j \ neq k) G_ (jk) (v_ (k) - v_ (j)) = \ sum _ (j \ neq k) G_ (jk) v_ (k) - \ sum _ (j \ neq k) G_ (jk) v_ (j) = G_ (kk) v_ (k) - \ sum _ (j \ neq k) G_ (jk) v_ (j)) where G k k (\ displaystyle G_ (kk)) is the sum of conductances connected to node k (\ displaystyle k). We note that the first term contributes linearly to the node k (\ displaystyle k) via G k k (\ displaystyle G_ (kk)), while the second term contributes linearly to each node j (\ displaystyle j) connected to the node k (\ displaystyle k) via G j k (\ displaystyle G_ (jk)) with a minus sign . If an independent current source / input i k (\ displaystyle i_ (k)) is also attached to node k (\ displaystyle k), the above expression is generalized to i k = G k k v k − ∑ j ≠ k G j k v j (\ displaystyle i_ (k) = G_ (kk) v_ (k) - \ sum _ (j \ neq k) G_ (jk) v_ (j)). It is readily to show that one can combine the above node - voltage equations for all N (\ displaystyle N) nodes, and write them down in the following matrix form </P>

Nodal analysis is primarily based on the application of