1

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

4

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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