[/img]

I'm trying to solve the above circuit using Kirchoff's laws but I don't seem to be having much luck. There is some fundamental problem I keep having, and I don't see what it is.

Lets call the current from R1, Ia.
Lets call the current from R2, I (which is point TOWARDS node B)
Lets call the current from R3, Ib.

Applying KCL at Node B gives us:

Ia + I - Ib = 0

Which means that: Ia + I = Ib

Applying KVL to the left-most loop:

0.5Ia - 10I = 6

Applying KVL to the right-most loop:

10 I + 12Ib = 18
10I + 12 (Ia + I) = 18
12Ia + 22I = 18

Does that seem right to you? To me it makes more logical sense for I to be heading AWAY from Node B, not towards it, and I think that's where I keep getting stuck (that's what the question specifies, that Ia and I are both entering Node B).

Any help is appreciated, thanks.

 

In most of the cases you don't know the currents sense in your circuit. You can make some assumptions, like you are trying to do in your post, but it is not necessary. You can choose any sense for the circuit currents, as long as you are consistent with the current sign in your equations. If the sense you chose is not the real one, when you solve the equations some currents will result as negative. Hence, they point backwards.

For example, after solving the system of equations you posted, the result is: Ia=2.38A, I=-0.481A, Ib=1.899A. I is negative, so yes, it points in the other direction.