The Best answer for What is the current through the 10 Ω resistor in the figure?
Use Thevenin's theorem to represent the 9v and 12v sources and their series resistors as a single voltage source in series with a single resistor.
Because those two sources are in series with the same value of resistor, the equivalent voltage is (1/2)*(9 + (-12)) = -1.5v.
The equivalent resistance is two 5 ohms in parallel, ie 2.5 ohms.
When that source is connected to the 3v source and its series resistor, the current is (3 + 1.5)v/(10 + 2.5) ohms = 0.36