What would happen if two voltage sources (one a DC battery and a dependent voltage source Vx) are in series?
A real voltage source with a voltage V and internal resistance Ri is a typical example of your arrangement. Its output voltage Vout depends on the load current IL (Vout = V - IL.Ri). So, you can think of this connection as two voltage sources (V and IL.Ri) in series... although IL.Ri is a voltage drop (a part of V that is lost).
A more correct example is the case when a true current source acts as a load; see for example Why is the voltage drop created by a current source added. The voltage drop created by the current source can be subtracted from the main voltage - Fig. 1,

Fig. 1. Opposing current source as a load.
or can be added to the input voltage - Fig. 2.

Fig. 2. Helping current source as a load.
The combination of the current source and resistor can be thought as another voltage source (Norton's arrangement) connected in series to the main voltage source.