A dietitian wishes to plan a meal around three foods. The meal is to include 7,600 units of vitamin A, 2,810 units of vitamin C, and 990 units of calcium. The number of units of the vitamins and calcium in each ounce of the foods is summarized in the following table.
  Food I  	Food II  	Food III
Vitamin A	400	1200	800
Vitamin C	110      	570     	340     
Calcium	90      	30     	60     

Determine the amount of each food the dietitian should include in the meal in order to meet the vitamin and calcium requirements. (Let x, y, and z denote the number of ounces of Food I, Food II, and Food III, respectively. If there is no solution, enter NO SOLUTION. If there are infinitely many solutions, express your answer in terms of the parameters t and/or s.)
(x, y, z) = 

 
  



We need to determine how many ounces of each food should be included in the meal such that the vitamin A, vitamin C, and calcium requirements are satisfied. The number of units of each nutrient in each food is summarized in the table.

### Let:
- \( x \) be the number of ounces of Food I,
- \( y \) be the number of ounces of Food II,
- \( z \) be the number of ounces of Food III.

We are given the following system of equations based on the requirements of the nutrients:

1. **Vitamin A Requirement**:
   \[
   400x + 1200y + 800z = 7600
   \]

2. **Vitamin C Requirement**:
   \[
   110x + 570y + 340z = 2810
   \]

3. **Calcium Requirement**:
   \[
   90x + 30y + 60z = 990
   \]

We can solve this system of equations to find \( x \), \( y \), and \( z \).

### Step 1: Solve the system of equations

Let me calculate the solution.The system of equations has infinitely many solutions, which can be expressed in terms of the parameter \( z \):

\[
x = 10 - \frac{z}{2}
\]
\[
y = 3 - \frac{z}{2}
\]

This means the dietitian can choose any value for \( z \), and the corresponding values of \( x \) and \( y \) will satisfy the vitamin and calcium requirements.

Thus, the solution is:

\[
(x, y, z) = \left( 10 - \frac{z}{2}, 3 - \frac{z}{2}, z \right)
\]

If you'd like further details on how to interpret this solution, feel free to ask!

### Follow-up questions:
1. What would be the values of \( x \), \( y \), and \( z \) if \( z = 2 \)?
2. Can the dietitian choose a negative value for \( z \)?
3. How does each food contribute to the nutrient intake?
4. What is the relationship between \( z \) and the ounces of food needed?
5. How does the choice of \( z \) impact the total amount of food required?

### Tip:
When working with infinitely many solutions, always check if the parameters (like \( z \)) make sense in the real-world context. For instance, negative quantities of food are usually not feasible.

A dietitian wishes to plan a meal around three foods. The meal is to include 7,600 units of vitamin A, 2,810 units of vitamin C, and 990 units of calcium. The number of units of the vitamins and calcium in each ounce of the foods is summarized in the following table. Determine the amount of each food the dietitian should include in the meal in order to meet the vitamin and calcium requirements. (Let x, y, and z denote the number of ounces of Food I, Food II, and Food III, respectively. If there is no solution, enter NO SOLUTION. If there are infinitely many solutions, express your answer in terms of the parameters t and/or s.)

This problem helps a dietitian plan a meal to meet specific vitamin A, vitamin C, and calcium requirements using a system of linear equations. Learn how to solve this system and express the solution parametrically. Suitable for students in Grades 10-12 studying linear algebra.

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