How do you solve and set up this problem?

Question

A chef wants to prepare a low-fat sodium meal using lean meat, roasted potatoes, and green beans. A 1-ounce serving of meat contains 50 Cal, 20 g of protein, and 16 mg of sodium. A 1-ounce serving of potatoes contains 9 Cal, 1 g of protein, and 3 mg of sodium. A 1-ounce serving of green beans contains 12 Cal, 2 g of protein, and 17 mg of sodium. If the chef wants to prepare a meal that contains 243 Cal, 73 g of protein, and 131 mg of sodium, how many ounces of each ingredient should be prepared?

Answer 1

Represent the quantities of calories, protein and sodium as an ordered set (cal,protein,sodium). If the quantities of meat, potatoes and beans are represented by m, p and b in ounces then the prepared meal will contain m(50,20,16)+p(9,1,3)+b(12,2,17)=(243,73,131). So 50m+9p+12b=243; 20m+p+2b=73; 16m+3p+17b=131. This is a system of equations with three variables and three equations, so it should be solvable.

Call the equations A, B and C.

3B-C: 60m+3p+6b-(16m+3p+17b)=219-131; 44m-11b=88, or 4m-b=8, so b=4m-8.

6B-A: 70m-3p=438-243=195, so 3p=70m-195.

We have b and p in terms of m, so we can substitute in any equation: let's pick B.

20m+(70m-195)/3+8m-16=73. Multiply through by 3:

60m+70m-195+24m-48=219; 154m=219+195+48=462, so m=3. Therefore b=4 and p=5.

So the balance is 3 oz of meat, 5 oz of potatoes, 4 oz of beans.