A glass of skim milk supplies 0.1 mg of iron, 8.5 g of protein, and 1 g of carbohydrates. A quarter 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 27.7 mg of iron, 178 g of protein, and 160 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 exactly this? (Round your answers to one decimal place.)
To determine how many glasses of skim milk, quarter-pound servings of meat, and two-slice servings of whole-grain bread will supply exactly 27.7 mg of iron, 178 g of protein, and 160 g of carbohydrates, we can set up a system of equations.
Let's denote the unknown variables as:
- x: number of glasses of skim milk
- y: number of quarter-pound servings of meat
- z: number of two-slice servings of whole-grain bread
Given the nutritional values of each food item, we can set up the following equations:
Equation 1: 0.1x + 3.4y + 2.2z = 27.7 (for iron)
Equation 2: 8.5x + 22y + 10z = 178 (for protein)
Equation 3: 1x + 20y + 12z = 160 (for carbohydrates)
Now, we can solve this system of equations using various methods such as substitution, elimination, or matrix operations.
Let's solve this system using elimination method:
Multiply Equation 1 by 10: 1x + 34y + 22z = 277
Multiply Equation 3 by 8.5: 8.5x + 170y + 102z = 1360
Now, we can eliminate the x variable by subtracting Equation 3 from Equation 2 and Equation 1 from Equation 3:
Equation 4: (1x + 34y + 22z) - (1x + 20y + 12z) = 277 - 160
Equation 5: (8.5x + 170y + 102z) - (8.5x + 22y + 10z) = 1360 - 178
Simplifying Equations 4 and 5:
Equation 4: 14y + 10z = 117
Equation 5: 148y + 92z = 1182
Now, we have a system of two equations in two variables (y and z). We can solve this system using the same method. Let's use the substitution method:
From Equation 4, we can rewrite it as:
y = (117 - 10z) / 14
Substitute this value of y into Equation 5:
148 * ((117 - 10z) / 14) + 92z = 1182
Simplifying the equation:
(148 * (117 - 10z) / 14) + 92z = 1182
(21028 - 1480z) / 14 + 92z = 1182
21028 - 1480z + 1288z = 16548
(1288 - 1480)z = 16548 - 21028
-192z = -4480
z = (-4480) / (-192)
z ≈ 23.33
Now, substitute the value of z back into Equation 4 to find y:
14y + 10(23.33) = 117
14y + 233.3 = 117
14y = 117 - 233.3
14y = -116.3
y = (-116.3) / 14
y ≈ -8.31
Since we cannot have negative values for servings of food, y ≈ -8.31 is not a valid solution.
Let's try the substitution method again, but we'll rearrange Equations 4 and 5 first:
Equation 4: 5y + 5z = 39 (multiply by 5)
Equation 5: 74y + 46z = 591 (divide by 2)
Now, solve Equation 4 for y:
y = (39 - 5z) / 5
Substitute this value of y into Equation 5:
74 * ((39 - 5z) / 5) + 46z = 591
Simplifying the equation:
(74 * (39 - 5z) / 5) + 46z = 591
(2874 - 370z) / 5 + 46z = 591
(2874 - 370z) + 230z = 2955
2874 - 140z = 2955
-140z = 81
z = 81 / -140
z ≈ -0.579
Again, we obtain a negative value for z, which is not valid.
Based on the given information and the calculations so far, it seems that there is no exact combination of servings of skim milk, lean red meat, and whole-grain bread that will supply exactly 27.7 mg of iron, 178 g of protein, and 160 g of carbohydrates. It is not possible to fulfill all the requirements simultaneously.
Please note that rounding the answers to one decimal place in the intermediate steps may introduce slight inaccuracies, which could contribute to the absence of an exact solution.