In Lyddie by Katherine Paterson, how did Lyddie become self-reliant? 

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Lyddie takes care of her whole family, and gets her own job when the tavern fires her.

Lyddie had to become self-reliant when her father left.  Her father went west to seek his fortune because he was not doing well at home.  The farm was failing.  That left Lyddie’s mother with four children to take care of, but she was barely capable of taking care of herself.  Lyddie acted, doing what had to be done.  She became the farmer and the parent.

An example of Lyddie’s self-reliance is breeding her cow to the Stevensons’ bull.  She does this behind her mother’s back, because her mother has gone “queer” in the head since their father left and barely knows what’s going on.  Lyddie knows that the family needs the money from the calves.

Early last summer, when Mama was having one of her spells and not paying much attention, Charlie had again sneaked the cow down the mountain to the Stevenses' place. As long as Lyddie could remember, long before their father had left, they had made use of the Stevenses' bull.  If their mother ever wondered about those calves that were born like miracles every spring, she never mentioned it. (Ch. 1) 

Lyddie took care of herself, and her family, because she had to.  When her mother left and hired her out to the tavern, she made the best of it.  When she was fired, she did not just wander around or ask someone for charity.  Lyddie continued her self-reliance and got herself a job.

 "I'm going to be a factory girl, Triphena."

"You what?"

"I'm free. She's set me free. I can do anything I want. I can go to Lowell and make real money to pay off the debt so I can go home." (Ch. 6) 

Lyddie does get a job, and she thrives at it.  She gets more and more machines assigned to her, even with the speed-up.  She also helps the new girls learn the ropes.  Lyddie defends herself against Mr. Marsden’s inappropriate advances.  She also defends Brigid, which gets her fired.  Even then Lyddie does not give up.  She decides to go to college!