One power supply for two systems?

I have a power supply with one 12V output. Can I use it to drive two systems each of which requires 12V by connecting the +12V output from the power supply to the +12V input of each system? All grounds will be connected. In this case, will the current from the power supply to each system be half the original?

No. The current supplied will be what each system demands - provided that it is capable of doing so.

If system 'A' needs 1 amp and 'B' needs 1/2 amp they take that amount of current as long as the 12V supply can supply 1.5 amps.

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Any disadvantages in using one power supply for two systems at the same time?

oceantable:
Any disadvantages in using one power supply for two systems?

Redundancy - if you loose the one power supply, both systems go down.

Loading - Using a single power supply for multiple systems loads down the supply (see post #2)

Noise - If you are using a single supply with no filtering, noise on the power rails on one system might propagate to the power rails on the other system.

oceantable:
Any disadvantages in using one power supply for two systems at the same time?

It depends entirely on what you mean by "system". A power supply's job is to produce power, and has nothing to do with systems, or how many there are. Load is load, and that is all it needs to know. However, it might be more convenient to have an individual power supply for each "system", due to factors that have nothing to do with power supply. It may also be more practical due to factors that do, like length of cable.

Nick_Pyner:
A power supply's job is to produce power, and has nothing to do with systems, or how many there are. Load is load, and that is all it needs to know.

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Power sources may only need to know how much load is being demanded by it, but overlooking the other complications (see post #4) of putting multiple loads in parallel is BAD DESIGN and can be fatal to the product/project.

I'm not saying the source needs to know information about the systems other than its load (although too much load can destroy the source - i.e. discharching a battery too much or too quickly); I'm looking at the problem from a high level design perspective (as OP is asking).