What is the solution to this circuit problem involving Kirchoff's Laws?

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The circuit problem involves two batteries and four resistors, with equations derived from Kirchhoff's Laws to find the currents I1, I2, I3, and I4. The user has set up equations based on the circuit's voltage and resistance but is struggling to obtain the correct values for I1 and I3. Substitutions were made to simplify the equations, but the results are still incorrect. The user is seeking assistance to resolve the discrepancies in their calculations. Clarification on the application of Kirchhoff's Laws and potential errors in the equations is needed for a solution.
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Homework Statement



The circuit in the figure is composed of two batteries (ε1 = 8 V and ε2 = 5 V) and four resistors (R1 = 110 Ω, R2 = 40 Ω, R3 = 50 Ω, and R4 = 50 Ω) as shown.

Image: See Attachment
OR: https://smart.physics.illinois.edu/images/phys102/Homework/03/cir06.gif

Homework Equations



I1=I4=I2+I3
E1-I1R1-I3R3-I4R4=0
E1-I1R1-I2R2-E2-I4R4=0

The Attempt at a Solution



First Equation: 8-110(I1)-50(I3)-50(I4)=0
Sub I4=I1 -> 8-160(I1)-50(I3)=0
Second Equation: 8-110(I1)-40(I2)-5-50(I4)=0
Sub I4=I1 -> 3-160(I1)-40(I2)=0
Sub I2=I1-I3 -> 160(I1)+40(I1)-40(I3)=3

Compare two equations:

160(I1)+50(I3)=8
200(I1)-40(I3)=3

The answers I got for I1 and I3 arent the correct answers. I've been working on this problem for awhile and am getting frustrated. Any help would be appreciated. Thanks, Doug
 

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