Need help with nodal analysis of a DC circuit.

Hi guys, first time poster and new forum member

Was hoping to get some help with nodal analysis of a circuit I have come across as an extension question in my class tutorials.

This is the circuit here: https://i.imgur.com/i4cY8B7.png

I'm looking to find the set of NODAL equations that mathematically describes the circuit. From there use Cramers Rule and Circuit Theory to solve for the voltages at the NODES 1 to 4 and then using these voltages at each node calculate the current for nodes 2 & 4.

I expect that the nodes 2 and 3 are not significant since they are outside the node? I.e. that I'd ignore the outside part?

Any help you guy can give would be greatly appreciated

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For example for node 1 I got the equation as:

-12 + (V1/2k Ohms) + (V4/2k Ohms) = 0

Node 4: -2 + (V6/2k Ohms) - (V5/4k Ohms) = 0

Where is V6 and V5 which you have mentioned in your node 4 equation!?
Do you have idea about supernode? Its like this.. because of 12V DC source, there forms a supernode.
u cant ignore V2 and V3
I am giving equation here..

at V1,V2: (V1-V3)/2K + V1/2K + (V1-V4)/2K +V2/1k + (V2-V4)/4K + (V2-V3)/5k = 0
auxiliary equation: V1-V2=12
at V4: (V4-V1) / 2K + 2m = 0
at V3: -Ia/5 + (V3-V1)/2K + (V3-V2)/5K = 0
and also : Ia = V1/2K

I have taken incoming current negative and outgoing current positive.
solve the equations and u should get your answers.
Let me know if u need more info.

Ok for the equation for node 4.

Why wouldn't it be (V4-V1)/2 + 2mA + (V4-V2)/4 = 0 ? Is it because we included it in the node 1,2 equation?

yes u are right. i missed (V4-V2)/4k in node4 equation!! Thanks for correcting me

Ok so for node 4:

I got 3V4/4 - V1/2 - V2/4 = -2mA ------- (Eq 1)

I got for Node 1/2:

(3V1/2 - V4/2 - V3/2 + I) + (29/20)V2 - V4/4 - V3/5 - I = 0

V1-V2=12 ------ (Eq 2)

Node 3:

Ia = V1/2k

Therefore 7V3 - 6V1 -2V2 = 0 ------(Eq 3)

Then by substituting Eq 2 into Eq 3 eliminates I

3V1/2 + 29V2/20 - 3V4/4 - 7V3/10 = 0 ------ (Eq 4)

Then the matrix I get and the solutions with Crammer's rule are here: https://i.imgur.com/5EEXbBv.jpg

i think your answers are right. voltage and current flow matches..

Ok quick question how did you find the directions for the current?

Current leaves from more positive voltage towards lower voltage.
for example: V1=7.143V so current through R4 is from V1 towards ground.
V4=0.476V and V2= -4.85V so current flows from V4 towards V2 in R5.
Its a convention followed.