https://file.aiquickdraw.com/mathbot/file/1726355723216fa6k5vt3.jpg

To solve the problem presented in the image, we can break it down into the following steps:

### Given Information:
- The dietitian is planning a meal of **carrots**, **corn**, and **beef**.
- The patient needs a total of:
  - 82.8 grams of carbohydrate,
  - 37.8 grams of protein, and
  - 17.6 grams of fat.

The table below shows the number of grams of carbohydrate, protein, and fat per 100 grams of each food:
| Food      | Carbohydrate (g) | Protein (g) | Fat (g) |
|-----------|------------------|-------------|---------|
| Carrots   | 8                | 1           | 0       |
| Corn      | 21               | 3           | 1       |
| Beef      | 1                | 22          | 12      |

Let \( x \), \( y \), and \( z \) represent the amounts (in grams) of carrots, corn, and beef respectively.

### System of Equations:
1. **Carbohydrate constraint**:
   \[
   8x + 21y + z = 82.8
   \]
2. **Protein constraint**:
   \[
   x + 3y + 22z = 37.8
   \]
3. **Fat constraint**:
   \[
   0x + y + 12z = 17.6
   \]

### Solving:
We have a system of three equations with three unknowns. Let's proceed to solve this system.

I'll calculate the values now.The solution to the system of equations is:

- \( x = 1.8 \) grams of carrots,
- \( y = 3.2 \) grams of corn, and
- \( z = 1.2 \) grams of beef.

This combination will meet the required amounts of carbohydrate, protein, and fat for the patient's meal.

Would you like any more details or clarifications?

Here are 5 related questions for further practice:
1. What would happen if the patient needed more protein, say 50 grams?
2. How would the solution change if carrots contained 12 grams of carbohydrates instead of 8?
3. What is the total weight of the meal for the patient using this food combination?
4. How could you reduce the amount of fat in the meal while maintaining the required protein?
5. Could you find a different combination of food that meets the same nutritional requirements?

**Tip:** When solving systems of equations, always check if you can simplify one of the equations first, as this often makes solving easier.

A dietitian is planning a meal of beef, corn, and carrots for a patient. The patient needs the meal to contain 82.8 grams of carbohydrate, 37.8 grams of protein, and 17.6 grams of fat. The table below shows the number of grams of carbohydrate, protein, and fat in 100 grams of each food. How much of each food should be used to ensure the patient gets the needed nutrients?

Learn how to solve a system of linear equations to plan a meal for a patient needing 82.8g carbohydrates, 37.8g protein, and 17.6g fat. This algebra problem involves carrots, corn, and beef, providing a step-by-step solution using key formulas. Suitable for high school students in Grades 10-12.

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