Node Voltage Method, how to resolve with Voltage and Current sources?
When you have a voltage source with one end at ground and you are using the node voltage method you immediately know the voltage at one node by inspection. You do need to be careful with the sign. Look in your textbook or course notes for an example.

When writing node voltage equations you are actually writing a KCL equation. It looks like you are adding all of the current leaving each node and setting the sum equal to zero, which is fine.

We won't solve your homework problems for you so I've just given some hints. You almost have the equations written correctly.