The topic of this problem is nodal analysis. In this problem we're going to work with circuits with independent current sources in them. The problem is to find the nodal equations for the circuit shown. The circuit that we have has two independent current sources in it, a 3 milliamp and a 6 milliamp source. It also has resistors 2k, 4k, 3k and another 4k on the right hand side of the circuit. So we're going to use nodal analysis, which means that the first step in our process is to identify the nodes and then write the nodal equations about each one of those nodes using Kirchhoff's current law so let's do that. The first step is to find the nodal or the nodes. The first step is to find the nodes. We have node one, we have node two, we have node three. To identify the top nodes and then we have a ground node which is a reference node, zero volt node at the bottom of the circuit. So we're going to sum the currents into each one of those nodes using Kirchhoff's Current Law and noting that Kirchhoff's Current Law can be applied whether you sum the currents into or out of each of the nodes. We just need to be consistent throughout the problem. And so in this problem we're going to solve the currents into each of those nodes. Starting at node one. So if we sum the currents in the node one, we have 3 milliamps flowing into node one that's our first term. We have current flowing up through the 2 kilohms from the bottom to the top starts at 0 volts goes to V sub 1 node of voltage at node 1 divided by 2k. So 0- V1 divided by 2K. We also have the current through the 4k resistor in the center of the circuit flowing from node 2 to node 1. And that's V2- V1 over 4k = 0. So, there's our first equation. We look at that equation, we see that there's two unknowns in that equation. Nodal voltages V1 and V2. We need to then continue writing our independent equations so that ultimately we have enough equations to solve for all of our unknowns. So looking at node two. Summing the currents into node two, we see we have six milliamps falling into node two from the bottom up to node two. We have the current flowing through the four kilohm from node one to node two through the four kilohm that's V1- V2 over 4 kilohms. And we have the current flowing from right to left through the 3 kilohm resistor so that's V3- V2 over 3 kilohms. That's all of our currents, so the sum is equal to 0. So when we look at our equations now, we have two equations and three unknowns, V1, V2, and V3. So we need our third known equation so that we have a set of equations which equals our sort of unknowns. We have three independent equations and three unknowns, V1, V2, and V3. So again, summing currents into node 3, we have -3 milliamps from the current source to the top of the circuit. We have the current flowing through the 3 kilohm resistor, V2 minus V3 divided by 3K, V3 divided by 2K, Sorry its, 3K. And we have the current flowing up through the 4 kilohm resistor on the right hand side of the circuit. And that is 0 minus V3 divided by 4k = 0. So that gives us a third equation which is independent of the top two. So we have three equations, three unknowns. You can solve this set of simultaneous equations using matrix analysis. You can solve it using elimination of variables, whatever approach is most comfortable to you. But if you solve this problem, then you can find each of the nodal voltages. Once you find those nodal voltages, you can find any other quantity in the circuit that is of interest to you.