To solve for three unknown variables, we need three independent equations. We can get these equations using Kirchhoff's Laws:
(This is sometimes called Kirchhoff's Current Law (KCL), or just the "junction rule".)
(This is sometimes called Kirchhoff's Voltage Law (KVL), or just the "loop rule".)
The first thing we need to do is to label the currents. In our previous examples it was obvious which way the currents would flow, but here it's not so clear. That's ok, though. We can still define a direction for each current, as shown in the diagram to the left. If we get the direction wrong, then the current will turn out to be negative.
Going counterclockwise around the right loop gives us the equation $$9-8-1I_3=0$$ which we can solve right away to find \(I_3=1\u{A}\).
Going clockwise around the left loop gives us the equation $$2-8-2I_1=0 \implies 2I_1=-6$$ which means \(I_1=-3\u{A}\). That means that a current of 3 amps will flow downward through the 2V battery.
Lastly, if we look at the junction at the top, $$I_1+I_2+I_3=0$$ This would have no solution if all the currents were positive, but of course we know that \(I_1\) is negative. Solving this for \(I_2\): $$I_2=-I_1-I_3=-(-3\u{A})-(1\u{A})=2\u{A}$$ so current will flow up the 8V battery.