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If I had a fire alarm system, or any thing for that matter, and I wanted to add battery back-up...
System normally powered by a 12V DC transformer, but if I added a 9V battery, would this cause problems?
I don't think that I've ever seen a fire alarm system that didn't already have a battery backup. And they have always been 12 Volt SLA batteries.
But, for other products...
It is very subjective. It all depends on what parts of the system need what voltages, and how much current they need. Some things may work, but probably not for very long.
For instance, if it were drawing 1/2 amp, then as soon as it switched over to the battery, the battery voltage will drop to about 6 Volts due to the internal resistance of the battery. Not many 12 Volt systems can run reliably at 6 Volts.