Finding Current Through Resistors in Figure

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The discussion focuses on calculating the current through resistors in a circuit with two batteries and three resistors. The equations set up for the circuit, including E2 - I3R3 - I2R2 = 0 and E1 - I1R1 - I2R2 = 0, are correct, but there are issues with algebraic manipulation and substitutions. The relationship I2 = I1 + I3 is crucial for simplifying the equations. Participants emphasize the importance of careful handling of signs and substitutions to avoid errors in the final current values. Properly solving the equations with these considerations will yield the correct current through each resistor.
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Determine the magnitude of the current through all the resistors shown in the figure

The batteries have emfs of E1 = 9.00V and E2 = 12.00V and the resistors have values of R1 = 15.8Ohm , R2 = 22.3Ohm, and R3 = 34.3Ohm.maybe I am just setting up my equations wrong?
E2 - I3R3 - I2R2 = 0
E1 - I1R1 - I2R2 = 0
I2 = I1 + I3

so i used one equation to solve for the other unknowns, and plugged in the i2 = i1 + i3 equation in the beginning. than used that equation to solve for the final one. maybe my algebra got screwed up along the way or something. it takes like a page working through the problem only to see that you didnt get it.so solving for I1 i tried
E2 -I1R1 - R2(E2 + I1R3)/(R3 + R2) = 0
which turned out to be .515 AI2:
E2 - R3(I2 + [I2R2/R1]) - I2R2 = 0
which turned out to be .199 AI3:
E1 - I3R3 - I3R2 - R2(E2 - I3R2)/(R1 + R2) = 0
which turned out to be .2299 A
sadly they're all wrong.
 

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E2 - I3R3 - I2R2 = 0
E1 - I1R1 - I2R2 = 0
I2 = I1 + I3
are correct. One must be careful with the substitutions and signs.

E2 - I3R3 - I2R2 = 0
E1 - I1R1 - I2R2 = 0

and one can use I2 = I1 + I3 to change I3 into I2 - I1.

With E2 - (I2 - I1) R3 - I2 R2 = 0
and E1 - I1 R1 - I2 R2 = 0,

one has two equations and two unknowns I1 and I2, and then find I3.
 

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