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.