Hello, Jim Baldwin;
Thank you for your reply. I guess the idea in your mind is the substitution method.
By the way, the followings give result:
Solve[eq[[{1, 3}]], {I1, I2}]
Solve[eq[[{1, 4}]], {I1, I2}]
Solve[eq[[{1, 5}]], {I1, I2}]
Solve[eq[[{2, 3}]], {I1, I2}]
Solve[eq[[{2, 4}]], {I1, I2}]
Solve[eq[[{2, 5}]], {I1, I2}]