Directions of currents in Kirchhoff's 2nd law problems

[User11037]

Homework Statement

Apply Kirchhoff's laws to find the current at point X in the circuit shown. What is the direction of the current?

Homework Equations

V = IR
Kirchhoff's 1st law: ∑Currents entering junction = ∑Currents leaving junctions
Kirchhoff's 2nd law: ∑EMFs in a loop = ∑PDs in a loop

The Attempt at a Solution

This problem is simple enough but my issue is how to tell initially which directions the currents are going in. My initial approach was as follows:

However, this leads to incorrect answers:

By Kirchoff's 1st law, I₂ = I₀ + I₁
By Kirchoff's 2nd law, the sum of the emfs in the top loop must equal the sum of the pds:
EMF = 4V
PDS:
V = IR, V = 20 x I₀
V = IR, V = 20 x I₁
So 20 x I₀ + 20 x I₁ = 4
Using I₂ = I₀ + I₁ from earlier,
20 x I₂ = 20 x I₀ + 20 x I₁
substituting this into 20 x I₀ + 20 x I₁ = 4
gives 20 x I₂ = 4
I₂ = 4 / 20 = 0.2A

Applying Kirchhoff's 2nd law to the big loop (ie ignoring middle resistor):
EMF = 10 + 4 = 14V
∑PDs = 80 x I₂ + 20 x I₀
80 x I₂ + 20 x I₀ = 14
80 x 0.2 + 20 x I₀ = 14
I₀ = -0.1A

I₂ = I₀ + I₁
I₁ = 0.2 -- 0.1 = 0.3 A which is not the correct answer.

Why should labelling the currents in this way not give the correct answer?

 

[SteamKing]

In these problems, if the assumed direction of the current is incorrect, the value obtained should just be the negative of the actual current magnitude. I haven't checked the set up of your equations, but there may be other problems there.

 

[NascentOxygen]

So 20 x I₀ + 20 x I₁ = 4

This equation is not correct. You have a sign wrong.

Once you have assigned directions to currents, you must pay close attention to polarities of associated potentials.