What are some reasons and explanations for why Lyddie had to leave her family to pay off debt?

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You answered the question nicely in the question.  Lyddie had to leave her family to pay off the family debt.  I would like to clarify though that Lyddie did not do the actual leaving of the family in the first place.  After the semi-bear attack in the first chapter, Lyddie's mother leaves the family farm to live with other relatives.  She takes all of the family with her except Lyddie and her brother.  Together, the two of them work the farm through the winter and care for the place.  Unfortunately, all of their hard work is to no avail, because the mom sold the farm.  The mother also sold Lyddie and her brother to indentured servitude.  Lyddie was sold to a local tavern and had to do all of the grunt work in order to help pay off the family debt.  The debt that I keep referring to is there because Lyddie's father presumably went off west in search of gold and has never returned.  He left the family with all of the financial burdens.  By the time that Lyddie decides to become a factory girl in the city, she has already been separated from her family (and not by her choice).  Lyddie's choice to work the factory is because she sees it as a chance to earn more money in a quicker fashion than she currently is able to.  

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