I think sentence 1 is the correct (can we), rather than sentence 2 (we can) but I am struggling to explain why this is.

  1. Only by working together can we make every one’s life better

  2. Only by working together we can make every one’s life better


This is a form of "negative inversion". The word "only" implies a negation: "Only by working together (and not any other way) ..."

This is a difficult and poorly understood aspect of English grammar. Some examples:

Never have I eaten sushi.

(compare "I have never eaten sushi")

Only on Monday do I wash my hair.

(Compare "I only wash my hair on Monday")

Your sentence could be rephrased without inversion:

We can only make everyone’s life better by working together

Only when the negative phrase starts the sentence is an inversion needed.

Wikipedia has further examples of negative inversion.

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