The main social system that informed Sargent Price's decisions to ultimately 'take care ' of his dog, Vicar was his time spent in the United States Marines. During Prices' time serving in the Marines, he and his fellow troops were tasked with shooting dogs on the streets.
As painful as it was for Price, a dog lover to have to do this, the Sargent knew that shooting the animals led them to have a quick death, as opposed to slowly succumbing to hunger or other unfortunate deaths.
When it came to his own dog, Vicar, Price recalled his time with the Marines and the dogs he had to put down during his service. In remembering this, Price knew that no matter how hard it was to let his own dog go, it was the most fair way to end his life.