Electrodynamics problem (circuit with two sources)

1. The problem statement, all variables and given/known data

In the circuit (see attachment) we have 2 sources. If we decrease the EMF of the first source (ε₁) by 1.5V, the current changes in every branch of the circuit. How does the EMF of the source ε₂ has to be changed, so that the current through the ε₁ source would be the same as previously?

2. Relevant equations

Kirchhoff's laws

3. The attempt at a solution

I guess what we need to do here is to find the dependency of ε₁ on ε₂. Is using the Kirchhoff's equations the best way here? I wrote down such eq:
[itex]
I_3R + I_1r_1 + I_5R = \epsilon_1
[/itex]
[itex]
3I_4R + I_6R + I_1r_1 = \epsilon_1
[/itex]
[itex]
I_2r_2 + I_6R - I_5R = \epsilon_2
[/itex]
[itex]
I_3 + I_4 = I_1
[/itex]
[itex]
I_5+I_6 = I_1
[/itex]
[itex]
I_3 = I_5 + I_2
[/itex]
[itex]
I_4 + I_2 = I_6
[/itex]
Here r₁, r₂ - internal resistances of the sources (since there is current flowing through the sources, they must have internal resistances, right?)
I₃ - top left resistor
I₅ - down left resistor
I₄ - top right resistor
I₆ - down right resistor
I₁ - ε₁ current
I₂ - ε₂ current
But couldn't solve the system :/
Maybe there is a better way to do this? It seems to me there should be something simpler than system of 8 equations?
Any help appreciated!

 

But then I₁ and I₂ are removed, too, and (I made a mistake in the first post here) it is not possible to find the dependency of I₁ on the other parameters (like ε₁, ε₂)? Or am I wrong and we are looking for something else?

 

Yes, thanks a lot, that was it!