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The problem asks for the proportions of servings of macaroni and cheese, broccoli, and chicken that would result in a meal with:

- **400 calories**
- **35 grams of protein**
- **10 grams of fiber**

The nutritional information per serving is given as follows:

| Nutrient         | Mac & Cheese | Broccoli | Chicken | Shells |
|------------------|--------------|----------|---------|--------|
| Calories         | 271          | 51       | 70      | 261    |
| Protein (g)      | 9            | 5.4      | 14      | 8      |
| Fiber (g)        | 2            | 5.2      | 0       | 5      |

### Variables for servings:
Let:
- \( x_1 \) = number of servings of **macaroni and cheese**
- \( x_2 \) = number of servings of **broccoli**
- \( x_3 \) = number of servings of **chicken**

### System of Equations:

1. **Calories equation**:
   \[
   271x_1 + 51x_2 + 70x_3 = 400
   \]

2. **Protein equation**:
   \[
   9x_1 + 5.4x_2 + 14x_3 = 35
   \]

3. **Fiber equation**:
   \[
   2x_1 + 5.2x_2 + 0x_3 = 10
   \]

I will now solve this system of equations for \( x_1 \), \( x_2 \), and \( x_3 \).The solution to the system of equations is:

- **Macaroni and cheese servings**: \( x_1 = 0.83 \) servings
- **Broccoli servings**: \( x_2 = 1.61 \) servings
- **Chicken servings**: \( x_3 = 1.35 \) servings

To meet the requirements of 400 calories, 35 grams of protein, and 10 grams of fiber, she should have approximately:
- 0.83 servings of macaroni and cheese,
- 1.61 servings of broccoli, and
- 1.35 servings of chicken.

Would you like further clarification or details?

Here are 5 related questions:
1. How would the proportions change if the calorie limit were increased to 500?
2. Can we substitute shells for some of the macaroni and cheese while still meeting the nutrition goals?
3. How does the fiber content affect the proportions in this solution?
4. What would happen if the protein requirement were reduced to 25 grams?
5. How would the solution differ if we eliminated broccoli?

**Tip:** To solve systems of equations with many variables, breaking them down and checking units early can simplify your process!

If she wants to limit her lunch to 400 calories but get 35 grams of protein and 10 grams of fiber, what proportions of servings of macaroni and cheese, broccoli, and chicken should she use?

Learn how to solve a system of linear equations to determine the proportions of macaroni and cheese, broccoli, and chicken servings that provide 400 calories, 35 grams of protein, and 10 grams of fiber. This problem is suitable for high school students in Grades 9-12 and covers key concepts in linear algebra and nutrition.

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