Confusion with Nodal Analysis w/ Sources

[RiceKernel]

Hi , I'm currently stuck on a problem that requires to be solved with nodal analysis.

I'm confused to what signs should I use when there's a voltage source in the loop.

The problem:

I already know the answer from the solutions but what I found is at V_0 :

(V0 - 60V)/12 + V0/12 + (V0 -24V)/6 = 0

This is horrendously incorrect but I can't understand why . I know current flows from higher potential to lower so if that was the case then my first term would be correct (matches solution) ; the 2nd term is also correct (V0-0V)/12 but the last term is incorrect. The correct term would be (VO - (-24V))/6 and I really don't know why? Can someone explain that to me? Even for the first term, I just got lucky by putting the correct sign but I want to know why and how does the sign vary.

Oh and if anyone wants the answer , V0= 3V. Power in resistors are easy to find once the currents and voltages are found.

Thanks,
GT

 

[Jony130]

Because 24V voltage source has a positive terminal connected to gnd and the minus terminal to 6 ohm resistor.
And this is why voltage at right side of a 6 ohm resistor is -24V ( 24 voltages lower than voltage at GND).

The reference point determines whether we have a positive or negative voltage.
See this example

 

[RiceKernel]

Does this apply regardless of the direction of the current(s) ?

 

[Jony130]

Does this apply regardless of the direction of the current(s) ?

In this case the current direction doesn't matter here. Because only the voltage source and the reference point (GND) determines the voltage polarity.
And this is why we have -24V and 60V regardless of the direction of the currents.