Difficult applied problems - for three equations. How do I create the equations?

Question

A dietician prepares meals for a patient under the guidance of a physician. The patient is prescribed a meal to have 800 calories, 55g of protien, and 220 mg of Vitamin C. The meal consists of roast beef - 300 cal, 20g protien, 0mg vitamin C; baked potato - 100 cal, 5g protien, 20mg vitamin C; and broccoli - 50 cal, 5g protien, 100 mg vitamin C.

How many servings of each food are needed in order to satisfy the doctor's orders?

Answer 1

Let r = servings of roast beef
Let p = servings of baked potatoes
Let b = servings of brocolli

(1) Calories
800 = 300r + 100p + 50b
16 = 6r + 2p + b
(2) Protein
55 = 20r + 5p + 5b
11 = 4r + p + b
(3) Vitamin C
220 = 0r + 20p + 100b
11 = p + 5b

From (3) p = 11 - 5b
(3) into (1):
(4)
16 = 6r + 2(11 - 5b) + b
16 = 6r + 22 - 10b + b
-6 = 6r - 9b
-2 = 2r - 3b

(3) into (2):
(5)
11 = 4r + 11 - 5b + b
0 = 4r - 4b
r = b

(5) into (4):
(6)
-2 = 2r - 3r
-2 = -r
r = 2

(6) into (5):
(7)
b = 2

(7) into (3):
(8)
11 = p + 5(2)
11 = p + 10
p = 1

Therefore, 2 servings each of roast beef and brocolli are required and 1 serving of baked potatoes is required.

Answer 2

Get the amount of calories first.

let r=roast beef, p=potato, b=broccoli

800=>300r+100p+50b

now protein

55=<20r+5p+5b

now vitamin C

220=<20p+100b

Now graph those regions and the region which fit everything is the possibility region. If you want them to equal exactly, take out the greater than/less than signs and find the point of intersection. You would eliminate a variable in each equation by modifying them so that the term was opposite, then adding the whole equation. From there, you would have a simple 2 variable system