what is the difference between "wohen mit" and "wohnen bei"?

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Answer

"Bei" is used in german when we want to indicate that we live/work etc "under" someone, for example "under someone's roof". As a result, "wohnen mit+Dativ" means "I live with someone in the same house", but the house is not necessarily his (i.e. the second person's). Otherwise, when someone lives in his parents' house, "er wohnt BEI seinen Eltern". The same applies to work. "Ich arbeite mit John", means that you work together in the same company/project etc. But "Ich arbeite bei John" means that you work in John's company. You also use "bei" when you want to name the company, for example "Ich arbeite BEI Siemens".

Answer

"Wohnen mit" applies to roommates or flatmates, or maybe some family members who share your house or apartment. Examples: 
- Er wohnt mit einigen Freunden zusammen in einer Wohngemeinschaft.
- Er wohnt mit seiner Frau und den Kindern in diesem Haus.

"Wohnen bei" expresses a kind of hierarchy. It refers either to the owner of the place, or at least somebody who actually pays for the place and allows you to live here as well. Examples: 
- Sie wohnt noch bei ihren Eltern.
- Ich wohne bei meinem Freund in dessen Haus, während meine Wohnung renoviert wird. 
- Während meines Urlaubs in Paris kann ich bei meinem Bruder wohnen.