I can appreciate that you want the correct answer. I want that for you too.
To figure if you provided more than half of a person's support, you must first determine the total support provided for that person. Total support includes amounts spent to provide food, lodging, clothing, education, medical and dental care, recreation, transportation, and similar necessities.
Generally, the amount of an item of support is the amount of the expense incurred in providing that item. For lodging, the amount of support is the fair rental value of the lodging.
Expenses that are not directly related to any one member of a household, such as the cost of food for the household, must be divided among the members of the household, so in your situation, you will need to take into account the $500/month your girlfriend receives for the chilren vs the true cost of the rent, utilities, total cost of food, personal and hygiene items for the children, along with any entertainment costs (including cable tv, toys, DVD's etc purchased solely for the children).
How many children does your girlfriend have, how much do you make, how much is your rent, your utilities, how much did you purchase for the children in the way of clothing, diapers, toys, etc etc etc....? more than $500/month overall? If its $500/month for 1 child, that's a little different than say $500/month for 5 children. This is a basic but tough concept. Is it beginning to make sense?
I promise I will not go away. I may be temporarily interrupted by things like work, life, etc, but I always come back to justanswer.
Please let me know if you have more questions.