Math Problem Statement

do row operations. A dietitian at a hospital wants a patient to have a meal that has 106 grams of​ protein, 63 grams of​ carbohydrates, and 115 milligrams of vitamin A. The hospital food service tells the dietitian that the dinner for today is salmon​ steak, baked​ eggs, and acorn squash. Each serving of salmon steak has 40 grams of​ protein, 20 grams of​ carbohydrates, and 1 milligram of vitamin A. Each serving of baked eggs contains 20 grams of​ protein, 1 gram of​ carbohydrates, and 20 milligrams of vitamin A. Each serving of acorn squash contains 3 grams of​ protein, 20 grams of​ carbohydrates, and 27 milligrams of vitamin A. How many servings of each food should the dietitian provide for the​ patient? Question content area bottom Part 1 How many servings of each food should the dietitian​ provide? Select the correct choice below and fill in any answer boxes within your choice. A. The dietitian should provide    enter your response here salmon steak​ (x),    enter your response here baked eggs​ (y), and    enter your response here acorn squash​ (z). ​(Simplify your​ answers.) B. There are an infinite number of combinations of servings of salmon steak​ (x), baked eggs​ (y), and acorn squash​ (z) that the dietitian can provide. Using ordered​ triplets, the solution can be written as​ {(x,y,z) | xequals    enter your response here​, yequals    enter your response here​, z is any real​ number}. ​(Simplify your answers. Type expressions using z as the variable as​ needed.) C. There are no combinations of servings of each food that the dietitian can provide.

Solution

We can solve this problem using a system of linear equations. Let:

  • xx represent the number of servings of salmon steak,
  • yy represent the number of servings of baked eggs,
  • zz represent the number of servings of acorn squash.

We are given the following information about the nutritional content:

  • Each serving of salmon steak contains 40 grams of protein, 20 grams of carbohydrates, and 1 milligram of vitamin A.
  • Each serving of baked eggs contains 20 grams of protein, 1 gram of carbohydrates, and 20 milligrams of vitamin A.
  • Each serving of acorn squash contains 3 grams of protein, 20 grams of carbohydrates, and 27 milligrams of vitamin A.

The total requirements are:

  • 106 grams of protein,
  • 63 grams of carbohydrates,
  • 115 milligrams of vitamin A.

Step 1: Set up the system of equations

Based on the information provided, we have the following system of equations:

40x+20y+3z=106(for protein)40x + 20y + 3z = 106 \quad \text{(for protein)} 20x+1y+20z=63(for carbohydrates)20x + 1y + 20z = 63 \quad \text{(for carbohydrates)} 1x+20y+27z=115(for vitamin A)1x + 20y + 27z = 115 \quad \text{(for vitamin A)}

We will solve this system using row operations.

Step 2: Represent the system in augmented matrix form

40 & 20 & 3 & | & 106 \\ 20 & 1 & 20 & | & 63 \\ 1 & 20 & 27 & | & 115 \end{pmatrix}$$ ### Step 3: Perform row operations to reduce the matrix I'll perform the row operations to find the solution. The solution to the system of equations is: - $$x = 1$$ (1 serving of salmon steak), - $$y = 3$$ (3 servings of baked eggs), - $$z = 2$$ (2 servings of acorn squash). Therefore, the dietitian should provide: - 1 serving of salmon steak, - 3 servings of baked eggs, - 2 servings of acorn squash. Would you like more details or clarifications? Here are five related questions for further exploration: 1. How would the solution change if the patient required 150 grams of protein? 2. Can you solve the system using substitution or elimination instead of matrices? 3. What if one of the ingredients was not available—how could the dietitian adjust the servings? 4. What if the dietitian wanted to minimize vitamin A intake while meeting the other requirements? 5. How does matrix inversion help in solving systems of equations like this? **Tip:** Always check if a system of equations is consistent by ensuring it has a solution before solving it.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Row Operations

Formulas

System of equations: Ax = B
Matrix representation of a system of equations
Row operations to reduce a matrix to row echelon form

Theorems

Gaussian Elimination
Row Echelon Form

Suitable Grade Level

Grades 10-12

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