Let's assume that the unit prices of Party A, Party B and Party C are a b c yuan respectively.

3a+7b+c=45

4a+8b+2c=78

So a+b+c = (4a+8b+2c)-(3a+7b+c) = 78-45 = 33 yuan.

Party A, Party B and Party C need 33 yuan to purchase 1 piece respectively.