Matrix Inverse Spreadsheet - Emergency!!!?

Question

A glass of skim milk supplies 0.1 mg of iron, 8.5 g of protein and 1 g of carbohydrates. Aquarter pound of lean red meat provides 3.4 mg of iron, 22 g of protein, and 20 g of carbohydrates. Two slices of whole-grain bread supply 2.2 mg of iron, 10 g of protein, and 12 g of carbohydrates. If a person on a special diet must have 12.1 mg of iron, 97 g of protein, and 70 g of carbohydrates, how many glasses of skim milk, how many quarter-pound servings of meat, and how many two-slice servings of whole-grain bread will supply this?

The answer to the question is 3/8lb of red meat, 6 slices of bread and 4 glasses of milk, but I need to know how to arrive at that answer.

Answer 1

.......skim.....meat....bread...total
iron.....0.1x + 3.4y + 2.2z = 12.1
prot.....8.5x + 22y + 10z = 97
carb .....1x + 20y + 12z = 70

enter the coefficients and the totals into a 3x4 matrix in your TI-83, ask for rref (reduced row echelon form). you get

1 0 0 4
0 1 0 3/2
0 0 1 3

which means 4 glasses of skim, 3/2*1/4 = 3/8 lbs red meat, and 3 2-slice servings of bread.

once you've got the system of equations, solve it by any convenient means.

Answer 2

In this problem, I'll use the following variable names:

X = glasses of milk
Y = pounds of meat
Z = slices of bread

A = milligrams of iron
B = grams of protein
C = grams of carbohydrates

Then, from the information provided, we know

A = 0.1*X + 3.4*4*Y + 2.2*(1/2)*Z
B = 8.5*X + 22*4*Y + 10*(1/2)*Z
C = 1*X + 20*4*Y + 12*(1/2)*Z

(Note the extra factors of 4 and 1/2 -- these are to make sure the answers are in pounds of meat and slices of bread, instead of in quarter-pounds of meat and pairs of slices of bread.)

Next, you want to find values of X, Y, and Z that will make A = 12.1, B = 97, and C = 70. To do this, you can convert the above system of equations into the following matrix equation (I apologize for the weird formatting -- the semicolons are supposed to denote vertical, or column, vectors):

[0.1, 3.4*4, 2.2*(1/2)]
[8.5, 22*4, 10*(1/2)] [X; Y; Z] = [12.1; 97; 70]
[1, 20*4, 12*(1/2)]

Then, you can invert the matrix and left-multiply the right-hand side of the equation by the resulting (inverted) matrix. Hopefully that will yield the correct results.