How many packages of each style should it prepare to use up existing​ supplies?

Question

A company produces three combinations of mixed vegetables that sell in​ 1-kg packages. Italian style combines 0.4 kg of​ zucchini, 0.4 kg of​ broccoli, and 0.2 kg of carrots. French style combines 0.5 kg of broccoli and 0.5 kg of carrots. Oriental style combines 0.2 kg of​ zucchini, 0.4 kg of​ broccoli, and 0.4 kg of carrots. The company has a stock of 19 comma 600 kg of​ zucchini, 37 comma 700 kg of​ broccoli, and 33 comma 700 kg of carrots. How many packages of each style should it prepare to use up existing​ supplies?

Answer 1

Let A, B and C represent the number of bags of Italian, French and Oriental mixes respectively. To use up all the stocks exactly:

0.4A + 0B + 0.2C = 19,600 (zucchini)

0.4A + 0.5B + 0.4C = 37,700 (broccoli)

0.2A + 0.5B + 0.4C = 33,700 (carrots)

Use the first equation to get C in terms of A

0.4A + 0.2C = 19600 -> 4A + 2C = 196000 -> 2C = 196000 - 4A -> C = 98000 - 2A

Replace C in the other two equations with 98000 - 2A, then likewise use the modified 2nd equation to get A in terms of C (or vice versa), then use that to replace in the 3rd equation, where you can then solve for A (or C). Once you have that value, work back thru the  2 conversion equations to derive the other two values.