Electrical – Find the voltage, Vx, and current, Ix

First off, mesh analysis would be simpler because you would only have one unknown (the current in the right mesh).

As you know, in nodal analysis you are summing currents at the nodes. Current into the nodes positive and current out of the nodes negative. The way you have defined the problem you should have 3 equations and three unknowns. Your first equations seems to do this correctly for node B. It kind of falls apart after that. For example, node A equation should just be .005 = (Va-Vb)/3000. You already know that Vc = 16 volts.

So simplifying the node A equation gives you what Andy aka said, then just substitute it back into your node B equation.

For part (b), the circuit to the left of the switch is disconnected so you can disregard it for any \$t>0\$.

The key insight for (b) is that the circuit is once again in DC steady state (all transients decay as \$t \rightarrow \infty\$).

Thus, the voltage across the inductor is \$0V\$, i.e., it can be replaced with an ideal wire.

Now, this leaves a simple current divider circuit so the inductor current is given by current division.