Circuit with coupled inductors, need help with mesh equations
You are using wrong sign of \$j3I_2\$ or \$j3I_1\$ in the wquations.
cause mutual flux comes from  other inductor  goes directly through the opposite direction of one's own flux. So your equations should be:
$$10\angle0 + j5I_1 - j9(I_1-I_2) + j3I_2 = 0 $$ 
$$28I_2 + j6I_2 -j3(I_1-I_2) -j9(I_1-I_2) + j3I_2=0$$