How can I completely shut off / inert my home alarm system?
Note that it would be dangerous to go without some compliant smoke alarm system - in the comments, @Anthony X indicates there are other smoke detectors / smoke alarms present.  

I'd say it's 99%+ that the battery and power supply are the only two sources for the system and if you unplug them it will be down.  You might as well unplug the phone line connection on the right too.  

Be sure to test the remaining smoke alarms, and make sure the system alarms as it should, and that there are still working smoke alarms in all the necessary areas.  At a minimum, you want one in every bedroom, and in the hallways outside of the bedromms, and make sure there's one on every floor of the house - including the basement.  

I would probably check out all the existing smoke detectors and replace them with something current.  Smoke alarms don't last forever, I'd rest easier if I knew what I have and had confidence that it works as it should.