1. 나한테 편지 써 줄거지?

    Are you going to me a letter?

=>  Are you going to write me a letter?

2. 물론이지. 너한테 이메일 써줄께.

    Of course. I'll write you an e-mail.  ✔

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3. 너 매일 매일 햄버거랑 피자 먹을 수 있겠다.

    You can eat hamburgers and pizza every day. 

=>  ​You'll eat pizza and hamburgers every day.

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• Corrected: The sentences are corrected grammatically.