What is the correct equation for calculating the currents in this circuit?

[roam]

Homework Statement

Use Kirchhoff's rules to find the currents I₁, I₂ and I₃ shown in the circuit below:

[PLAIN]http://img811.imageshack.us/img811/3640/13608736.gif

The Attempt at a Solution

Kirchhoff's junction law gives: I₁+I₂-I₃=0

I have calculated the voltage rise/drop in the left inner loop containing the 10-V battery:

10-10I₁+20I₂=0

Also in the right inner loop containing the 30-V battery:

30-20I₂-20I₃=0

So we have the 3 equations:

I₁+I₂-I₃=0
-10I₁+20I₂=-10
-20I₂-20I₃=-30

We can pust this in a matrix to solve for the three currents:

$$\begin{pmatrix}1&1&-1&0\\-10&20&0&-10\\0&-20&-20&-30\end{pmatrix}$$

which row reduces to

$$\begin{pmatrix}1&0&0&2.2\\0&1&0&-0.3\\0&0&1&1.8\end{pmatrix}$$

So I₁= 2.2, I₂=-0.3 and I₃=1.8, but this is wrong. The correct answer must be I₁=4, I₂=1.5, I₃=5.5 Amps.

So, what's wrong with my calculations?

 

[rl.bhat]

-20I2-20I3=-30

This equation is wrong. Check it.

 

[roam]

-20I2-20I3=-30

This equation is wrong. Check it.

So what is the correct equation for that? I have checked it, if the direction assigned to the loop is clockwise, we have -20I₂-20I₃+30=0. What's the problem?

 

[rl.bhat]

So what is the correct equation for that? I have checked it, if the direction assigned to the loop is clockwise, we have -20I₂-20I₃+30=0. What's the problem?

for I₃ there is no resistance for fall of potential.