Sure you can swap out your logic board with 4 GB of RAM
* 13” MacBook Air Logic board, 1.86GHz, 4GB: Apple P/N 661-5798 - Aprox $180 USD
* 13” MacBook Air Logic board, 2.13GHz, 4GB: Apple P/N 661-5734 - Aprox $200 USD
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Here’s the guide [https://www.ifixit.com/Guide/MacBook+Air+13-Inch+Late+2010+Logic+Board+Replacement/4511]
The OS is on the SSD drive which you’ll transfer to the replacement logic board so you won’t loose anything or need to re-install your OS.
But with what I’ve said here I don’t think this makes much sense given the age of the system and the cost as well as availability of the part. I would strongly recommend looking for a newer used MacBook Air or Pro system as being a smarter investment.
Sure you can swap out your logic board with 4 GB of RAM
* 13” MacBook Air Logic board, 1.86GHz, 4GB: Apple P/N 661-5798 - Aprox $180 USD
* 13” MacBook Air Logic board, 2.13GHz, 4GB: Apple P/N 661-5734 - Aprox $200 USD
The OS is on the SSD drive which you’ll transfer to the replacement logic board so you won’t loose anything or need to re-install your OS.
But with what I’ve said here I don’t think this makes much sense given the age of the system and the cost as well as availability of the part. I would strongly recommend looking for a newer used MacBook Air or Pro system as being a smarter investment.