Nodal Analysis - No Resistor

I am having problems with this circuit. The circuit has 3 nodes - v1,v2 and v3. Solve for the voltage across the 3 ohm resistor using nodal analysis. the 3 equations I get are as follows:
1) (v1-v3)/6-2+(v1-v2)/4+v1/0=0
2) (v2-v1)/4+v2/0+(v2-v3)/2=0
3) (v3-v1)/6+2+(v3-v2)/2+v3/3=0
This is obviously wrong. The 1st equation has v1/0 and the second equation has v2/0. How do I get rid of them?
Can somebody help with this please?

 

Just to clarify your schematic. Is the earth [0V] connected to both the bottom "rung" and the center of the first "rung"?

 

Simply ignore V3
Write the nodal equation only for V1 and V2

 

yes it is.

 

Yes it is.

 

Could you clarify please?

 

Is there no current through v3? Ignoring v3 I get:
1) (v1-v2)/6-2+v1/0=0
2) (v2-v1)/6+2+v2/3=0

I still have a divide by 0 in 1). Can v1/0 be replaced by -v2/3?

 

It would be useful for you to re-draw the schematic taking account of which elements are connected to the common [0V] rail. You'll find that apart from the common node there is only one other node.

 

Ok so the diagram look like this



And we only have one unknown node voltage V1.