The table on the right shows the​ calories, sodium, and protein in one cup of various kinds of soup. How many cups of each kind of soup should be mixed together to produce 10 servings of​ soup, each of which provides 219 ​calories, 1 comma 116 milligrams of​ sodium, and 12.6 grams of​ protein?  What is the serving size​ (in cups)?
Chicken Noodle
Chicken​ & Rice
Chicken Rotini
Calories
Sodium​ (mg)
Protein​ (g)
100
780
7
 
130
730
9
 
140
530
6
 
 
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Part 1
To produce 10 servings of​ soup, each of which provides 219 ​calories, 1 comma 116 milligrams of​ sodium, and 12.6 grams of​ protein, mix 
  
enter your response here cups of Chicken​ Noodle, 
  
enter your response here cups of Chicken​ & Rice, and 
  
enter your response here cups of Chicken Rotini.

To solve this problem, we can set up a system of equations based on the number of cups of each kind of soup (Chicken Noodle, Chicken & Rice, and Chicken Rotini) needed to meet the total requirement of calories, sodium, and protein for 10 servings. 

Let's define the following variables:
- \( x \): cups of Chicken Noodle soup
- \( y \): cups of Chicken & Rice soup
- \( z \): cups of Chicken Rotini soup

Each serving provides:
- **Calories**: 219 calories per serving, so for 10 servings, the total calorie requirement is \( 219 \times 10 = 2190 \) calories.
- **Sodium**: 1,116 mg of sodium per serving, so for 10 servings, the total sodium requirement is \( 1116 \times 10 = 11160 \) mg.
- **Protein**: 12.6 grams of protein per serving, so for 10 servings, the total protein requirement is \( 12.6 \times 10 = 126 \) grams.

Next, let's break down the information about each type of soup:
- Chicken Noodle: 100 calories, 780 mg sodium, 7 g protein per cup.
- Chicken & Rice: 130 calories, 730 mg sodium, 9 g protein per cup.
- Chicken Rotini: 140 calories, 530 mg sodium, 6 g protein per cup.

We can now set up the system of equations based on these three conditions:

1. **Calories equation**:
\[
100x + 130y + 140z = 2190
\]

2. **Sodium equation**:
\[
780x + 730y + 530z = 11160
\]

3. **Protein equation**:
\[
7x + 9y + 6z = 126
\]

### Solving the system of equations

Now, let's solve this system to find the values of \( x \), \( y \), and \( z \). I will calculate this for you.The solution to the system of equations is:

- \( x = 3 \) cups of Chicken Noodle soup,
- \( y = 7 \) cups of Chicken & Rice soup,
- \( z = 7 \) cups of Chicken Rotini soup.

Thus, to produce 10 servings of soup, you should mix 3 cups of Chicken Noodle, 7 cups of Chicken & Rice, and 7 cups of Chicken Rotini.

### Serving Size:
Since these amounts give 10 servings, the serving size in cups is the total number of cups divided by 10:

\[
\text{Total cups} = 3 + 7 + 7 = 17 \text{ cups}
\]
\[
\text{Serving size} = \frac{17}{10} = 1.7 \text{ cups per serving.}
\]

Would you like any further details or clarification?

Here are 5 related questions to expand on this:
1. How would the amounts change if each serving needed 250 calories instead of 219?
2. Can you calculate the number of servings if you only have 2 cups of each soup available?
3. How much sodium and protein would be provided in one serving using these amounts of soup?
4. How would the solution change if you added a fourth kind of soup to the mix?
5. What would the total calorie content be if you used double the amount of Chicken Noodle soup?

**Tip:** When solving systems of equations in practical scenarios, ensure that all units (like mg, grams, etc.) are consistent across the problem for accurate results.

The table on the right shows the​ calories, sodium, and protein in one cup of various kinds of soup. How many cups of each kind of soup should be mixed together to produce 10 servings of​ soup, each of which provides 219 ​calories, 1,116 milligrams of​ sodium, and 12.6 grams of​ protein? What is the serving size​ (in cups)?

Learn how to mix Chicken Noodle, Chicken & Rice, and Chicken Rotini soups to meet specific nutritional requirements of 219 calories, 1,116 mg sodium, and 12.6 g protein per serving for 10 servings. This problem involves solving a system of linear equations. Suitable for Grades 9-12.

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