I have this circuit 
I have to find \$V_a\$ and \$V_b\$, when I tried to do so, using node analysis I had the Equation \$\frac{V_a}{6} + \frac{V_a-12}{3}+\frac{V_a-V_b}{5} – 6 – 2 = 0\$ and \$\frac{V_b}{22}+\frac{V_b-20}{4}+\frac{V_b-V_a}{5} +2-3=0\$. Solving it simultaneously, I got \$V_a = 23.29 \text{V}\$ and \$V_b=21.51 \text{V}\$. When I simulated it using Circuit simulator applet, the simulation says that \$V_a = 24.4 \text{V}\$ and \$V_b\$ is \$25.4 \text{V}\$. Can someone point out where did I go wrong? Thanks!
Best Answer
There are several mistakes. But you still did pretty good. Let's look at the schematic:
simulate this circuit – Schematic created using CircuitLab
Now all of the important nodes have been labeled and a new unknown current identified.
Your first equation should be:
$$\begin{align*} \frac{V_a}{R_1}+\frac{V_a}{R_2}+\frac{V_a}{R_3}&=\frac{V_1}{R_1}+\frac{0\:\text{V}}{R_2}+\frac{V_b}{R_3}+I_1+I_2\\\\\therefore\\\\\frac{V_a-V_1}{R_1}+\frac{V_a-0\:\text{V}}{R_2}+\frac{V_a-V_b}{R_3}-I_1-I_2&=0\:\text{A} \end{align*}$$
And you got that one right! So there's no problem, yet.
Your second equation should be:
$$\begin{align*} \frac{V_b}{R_3}+\frac{V_b}{R_4}+\frac{V_b}{R_5}+I_1&=\frac{V_a}{R_3}+\frac{V_c}{R_4}+\frac{V_c+V_2}{R_5}+I_3\\\\\therefore\\\\\frac{V_b-V_a}{R_3}+\frac{V_b-V_c}{R_4}+\frac{V_b-V_c-V_2}{R_5}+I_1-I_3&=0\:\text{A} \end{align*}$$
Note that this is quite different from yours. One of the differences is that you used \$22\:\Omega\$ instead of \$R_4=12\:\Omega\$, I think. But actually there are more serious errors, as you can see. So this is really minor, by comparison.
You need two more equations. One of them is very simple:
$$\begin{align*} \frac{V_c+V_2}{R_5}&=\frac{V_b}{R_5}+I_4\\\\\therefore\\\\\frac{V_c+V_2-V_b}{R_5}-I_4&=0\:\text{A} \end{align*}$$
The final one is:
$$\begin{align*} \frac{V_c}{R_4}+\frac{V_c}{R_6}+I_4&=\frac{V_b}{R_4}+\frac{0\:\text{V}}{R_6}\\\\\therefore\\\\\frac{V_c-V_b}{R_4}+\frac{V_c-0\:\text{V}}{R_6}+I_4&=0\:\text{A} \end{align*}$$
That's it. From these four you can work out all of \$V_a\$, \$V_b\$, \$V_c\$, and \$I_4\$.
Took me less than 10 minutes to write this out.