# (a) Show that the system of linear equations x + 4y + 7z = −3 2x + 6y + 5z = 10 8x + 9y

(a) Show that the system of linear equations x + 4y + 7z = −3 2x + 6y + 5z = 10 8x + 9y + 5z = 1 can be written as Ax = b. (b) A small firm manufactures and sells litre cartons of non-alcoholic cocktails, ‘The Caribbean’ and ‘Mr Fruity’, which sell for \$1 and \$1.25, respectively. Each is made by mixing fresh orange, pineapple and apple juices in different proportions. The Caribbean consists of 1 part orange, 6 parts pineapple and 1 part apple. Mr Fruity consists of 2 parts orange, 3 parts pineapple and 1 part apple. The firm can buy up to 300 litres of orange juice, up to 1125 litres of pineapple juice and up to 195 litres of apple juice each week at a cost of \$0.72, \$0.64 and \$0.48 per litre, respectively. Find the number of cartons of ‘The Caribbean’ and ‘Mr Fruity’ that the firm should produce to maximize profits. You may assume that non-alcoholic cocktails are so popular that the firm can sell all that it produces.