Imo, the most straightforward of solving all these sorts of circuit problems is to consider the voltage gain/drop across every loop.

Meaning, pick a point in the circuit (starting above the 40V battery let's say), then consider the voltage gain/drop as you move through the 40V battery, through the 80ohm resistor (v = i_1 r_1), and back through the 200ohm resistor (v = i_2 r_2). Since you've made a full loop, the sum of these voltage gains/drops must add to 0. This gives you one equation with two unknowns (the two currents).

Then, you can similarly write down some more equations by considering other round-trip loops of voltage gain/drop and setting them equal to 0. Once you've written the equations for loops that pass through every component in your circuit, just use your favorite way of solving linear systems of equations.