The topic of this problem is nodal analysis. The problem is to determine the nodal equations for each node in the circuits drawn. In the circuits that we have, we have a 12 volt source on the left hand side of the circuit and a series of parallel or resisters through out the rest of the circuit. So when we're doing nodal analysis the first thing we do is we identify all the nodes in the circuit. Remember that a node is a connection between two elements in the circuit. So we have a node 1 at the left hand side of the circuit, we have a node 2 at the top we have a node 3 at the top, a node 4 at the top, and we always have kind of our ground node at the bottom of the circuit which we know the voltage for already, it's 0V. So whenever we're doing nodal analysis we have to first of all acknowledge that when we're doing nodal analysis that we're finding the nodal voltage and we're going to use these equations these nodal equations ultimately to find the nodal voltages associated with a circuit. In the problem we're not going to carry it all the way to the point where we find those voltages V1, V2, V3 and V4, but we're going to set the equation so that it can be done. We noticed that if we were to go far as to answer the nodal voltages to find the nodal voltages. That would ultimately give us ability to find al the currents in the circuit because we know the nodal voltages and we know all voltage drops across the resistive elements in the circuit. If we know the voltage drops across the resist of elements, then we can find the current and then we could find the power if we want to associate with the resistors and the voltage source as well. So we going to go ahead and setup the equations for this problem. So starting with node 1, looking at node 1. We find that node 1 has two currents associated with it. If we're going to sum the currents into the node, we're going to use that as our approach in this problem. So we're going to do KCL, Kirchhoff's Current Law and we're going to sum the currents into the nodes. Looking at node 1 we see that we have a voltage source. On the left hand side of the circuit which is connected through node 1 and we have a 9 kilo Ohm resistor at the top on the left hand side of the circuit which also is a current associated with node 1. So, we would have to find the current through the voltage source as well as current through the 9 kilo Ohm resistor. But again, remembering what we're after with nodal analysis, is we're trying to determine the nodal voltages. So in this first node, we don't have to sum the currents because we already know what the nodal voltage is, V1 = 12 V. We know that because it has a voltage between the node 1 and the ground node at the bottom of the circuit. Giving us a reference point and hopefully letting us know that v sub 1 is equal to 12 volts. So if we move on to node 2 a more, A regular type of node for summing current into a node for nodal analysis. So we have the current flowing through the 9 kilo Ohm resister from node 1 to node 2. We have the current flowing through the 6 kilo Ohm resister from the ground node to node 2. And we also have the current flowing through the 3 kilo Ohm resister at the top of the circuit. From node three to node two. So we're going to sum those up, starting with the 9 kilo Ohm resistor. It's going to be the current flowing into that node is going to be V1- V2 / 9K. We're next going to add the current through the 6 kilo Ohm resistor, flowing up through it from node 0 to node 2. It's going to be V0, which is 0- V2 / 6K. And we have the current flowing right to left through the 3 kilo Ohm at the top of the circuit and that's going to be V3- V2 / 3 kilo Ohm, and that's equal to 0. So that's our second equation and we can already that we have three unknowns with the first equation gives us one of those unknowns, so ultimately we have two unknowns left based on those two equations. But we're continue with the circuit and we're going to add these other unknowns as well. So we're going to look at node 3 and again summing the currents into node 3. First all through the 3 kilo Ohm resistor at the top of the circuit is going to be V2- V3 / 3 kilo Ohms. The current flowing up through the four kilo Ohm resistor from bottom to top is going to be zero for the ground node- V3 / 4k and we have the current flowing through the 9k on the right hand side of the circuit at the top. (V4- V3) / 9k = 0. So that gives us a third independent equation. And continuing on to node 4 on the right hand side of the circuit, we have two currents flowing in to node 4. First of all through the 9k resistor at the top it's going to be V3- V4 / 9 K + the current flowing up through the 3k resistor on the right hand side of the circuit which is 0- V4. It's a ground node- the voltage at node four / 3k = 0. So now we have these four equations. And we notice if we survey those equations that we have four unknowns, V1 through V4. And so, we can use these equations and solve for each of our unknowns using simultaneous solutions for these four equations.