On one day at a local minigolf course, there were 414 customers who paid a total of $3,780. If the cost for a child is $9 per game and the cost for an adult is $12 per game, write a system of equations to model this scenario, where x represents the number of children and y represents the number of adults who played that day.

x + y = 414
12x + 9y = 3780
x + y = 414
9x + 12y = 3780
x + y = 3780
12x + 9y = 414
x + y = 3780
9x + 12y = 414