Finding Current Through Resistors in Figure

[khoaz]

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.

 

[Astronuc]

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.