Example 4: (Homework) Find all node voltages with respect to the top-left corner treated as reference node:

$V_1=12 V$, $V_2=6 V$, $R_1=3 \Omega$, $R_2=8 \Omega$, $R_3=6 \Omega$, $R_4=4\Omega$.

Solution: First assume the bottom node $V_0=0V$ is reference point, and we have node voltages $V_1=12 V$ (left), $V_2$ (middle) and $V_3=-6V$ (right). Applying KCL to $V_2$, we get

$$\frac{V_2-(-6)}{6}+\frac{V_2-12}{3}+\frac{V_2}{8}=0$$

Solving for $V_2$ we get $V_2=24/5$. Now treating $V_1$ as reference point (ground), we have $V_1=12-12=0$, $V_2=24/5-12=-36/5$, $V_3=-6-12=-18$, $V_0=0-12=-12$.