A dietitian at a hospital wants a patient to have a meal that has 147 grams of​ protein, 88 grams of​ carbohydrates, and 158.5 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 30 grams of​ protein, 10 grams of​ carbohydrates, and 1 milligram of vitamin A.  Each serving of baked eggs contains 20 grams of​ protein, 1 gram of​ carbohydrates, and 15 milligrams of vitamin A.  Each serving of acorn squash contains 4 grams of​ protein, 20 grams of​ carbohydrates, and 37 milligrams of vitamin A.  How many servings of each food should the dietitian provide for the​ patient?
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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.


This is a system of equations problem where we need to determine how many servings of each food the dietitian should provide to meet the required amounts of protein, carbohydrates, and vitamin A.

Let:
- \( x \) be the number of servings of salmon steak,
- \( y \) be the number of servings of baked eggs,
- \( z \) be the number of servings of acorn squash.

From the problem, we can set up the following system of equations based on the nutritional content of each food:

1. **Protein equation**:
   \[
   30x + 20y + 4z = 147 \quad \text{(total grams of protein)}
   \]
   
2. **Carbohydrate equation**:
   \[
   10x + 1y + 20z = 88 \quad \text{(total grams of carbohydrates)}
   \]
   
3. **Vitamin A equation**:
   \[
   1x + 15y + 37z = 158.5 \quad \text{(total milligrams of vitamin A)}
   \]

Now, I will solve this system of equations.The dietitian should provide the following servings of each food to meet the nutritional requirements:

- 2.5 servings of salmon steak,
- 3 servings of baked eggs,
- 3 servings of acorn squash.

Would you like more details on the solution or further clarification?

Here are 5 related questions to expand your understanding:
1. How can you verify that these servings meet the exact nutritional requirements?
2. How would the equations change if the nutrient requirements were different?
3. What methods can you use to solve similar systems of equations?
4. What if the servings of a particular food needed to be whole numbers?
5. How would the solution change if a fourth type of food were added?

**Tip:** When solving a system of equations, always double-check the units (grams, milligrams, etc.) to ensure the calculations are accurate.

A dietitian at a hospital wants a patient to have a meal that has 147 grams of protein, 88 grams of carbohydrates, and 158.5 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 30 grams of protein, 10 grams of carbohydrates, and 1 milligram of vitamin A. Each serving of baked eggs contains 20 grams of protein, 1 gram of carbohydrates, and 15 milligrams of vitamin A. Each serving of acorn squash contains 4 grams of protein, 20 grams of carbohydrates, and 37 milligrams of vitamin A. How many servings of each food should the dietitian provide for the patient?

Learn how to solve a system of equations involving protein, carbohydrates, and vitamin A requirements for a hospital patient's meal. The problem uses three equations to determine the number of servings of salmon steak, baked eggs, and acorn squash. Suitable for Grades 9-12 students learning algebraic systems.

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