You did not mention the state in which you live and received the ticket in so we cannot state specific state laws. We can say in general that if the passenger does not pay their seat belt ticket it would not normally affect you since the ticket was written to her name and thus it would affect her.

As for taking her to court to pay your ticket, you could discuss this with a lawyer however we doubt that you would be able to do this. If your state law is that the driver must make sure that all passengers wear their safety belts and you neglected to do so then you would be held at fault for your own ticket.

If there were extenuating circumstances, such as you asked her to buckle up and she did but then undid her seat belt and refused to put it back on after you were already on the roadway then you may have be able to take issue with the citation and thus can try to fight it in court. If you lose and have to pay in court then again you would need to seek legal advice with someone familiar with this type of law in your state to see if you could go against your passenger for any reason. 

To find out about the fine for your seat belt violation ticket contact the court listed on the ticket. The clerk of the court should be able to tell you about the penalties associated with your ticket and if there are any points associated with this type of offense.

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