Learn Hardcore Korean: bangbeobeul bakkulkkayo?

It’s a polite, inclusive proposal: “Shall we…?” It can also ask for the listener’s opinion or a guess: “Do you think…?” In this sentence it’s a suggestion: “Shall we change (it)?”

Who is the subject? Does it mean “Shall I” or “Shall we”?

Korean often omits the subject. With -(으)ㄹ까요?, the default reading is inclusive “we.” Context can shift it:

제가 방법을 바꿀까요? = “Shall I change the method?”
우리 방법을 바꿀까요? = “Shall we change the method?”

Why is it 을 after 방법 and not 를?

을/를 marks the direct object. Use 을 after a final consonant (받침) and 를 after a vowel. 방법 ends with ㅂ, so 방법을 is correct. If the noun ended in a vowel, you’d use 를 (e.g., 계획을/차를).

How is 바꾸다 changed to 바꿀까요? exactly?

Dictionary form: 바꾸다 (stem: 바꾸-)
Add the proposal/guess ending -ㄹ까요? (since the stem ends in a vowel)
Result: 바꾸 + ㄹ까요 → 바꿀까요? Compare: 가다 → 갈까요?, 먹다 → 먹을까요? (consonant-ending stem takes -을까요?)

What’s the difference between 바꾸다 and 바뀌다?