The two answers that best explain the saying "a rich man's war but a poor man's fight" are:
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Wealthy slave owners in the south did not fight in the war, but poor men were drafted.
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Rich men did not care about the war, only poor men did.
These responses indicate that while the wealthier individuals may have had a vested interest in the outcomes of the war (such as maintaining slavery and their economic interests), it was often the poorer population who were the ones actually fighting in it, either through conscription or lack of alternatives.