A botanist can purchase plant food of four different types I II III and IV Each food comes in the same size bag and the following table summarizes the number of grams of each of three nutrients that each bag contains Food grams I II III IV Nutrient A 5 5 10 5 Nutrient B 10 5 30 10 Nutrient C 5 15 10 25 The botanist wants to use a food that has these nutrients in a different proportion and determines that he will need a total of 9 500 g of A 21 500 g of B and 16 500 g of C Find the number of bags of each type of food that should be ordered Let x R Type I bags y Type II bags z Type III bags and w Type IV bags If there are infinitely many solutions express your answers in terms of w as in Example 3