A​ nutritionist, working for​ NASA, must meet certain minimum nutritional requirements and yet keep the weight of the food at a minimum. He is considering a combination of two​ foods, which are packaged in tubes. Each tube of food A contains 4 units of​ protein, 2 units of​ carbohydrates, and 2 units of fat and weighs 2 pounds. Each tube of food B contains 3 units of​ protein, 6 units of​ carbohydrates, and 1 unit of fat and weighs 2 pounds. The requirement calls for 54 units of​ protein, 36 units of​ carbohydrates, and 22 units of fat. How many tubes of each food should be supplied to the​ astronauts?
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Part 1
The number of tubes of food A is 
  
enter your response here.

This is a system of linear equations problem. We can define the number of tubes of food A as \( x \) and the number of tubes of food B as \( y \). Using the given nutritional information, we set up the following system of equations based on the nutritional requirements:

### Equations:
1. **Protein requirement:**
   \[
   4x + 3y = 54
   \]
   (4 units of protein in each tube of A and 3 units in each tube of B, and the total required is 54 units).

2. **Carbohydrates requirement:**
   \[
   2x + 6y = 36
   \]
   (2 units of carbohydrates in each tube of A and 6 units in each tube of B, and the total required is 36 units).

3. **Fat requirement:**
   \[
   2x + y = 22
   \]
   (2 units of fat in each tube of A and 1 unit in each tube of B, and the total required is 22 units).

### Solve the system of equations:

Let's solve this step by step.

#### Step 1: Solve the second equation for \( y \):
\[
2x + 6y = 36 \quad \Rightarrow \quad x + 3y = 18 \quad \Rightarrow \quad 3y = 18 - x \quad \Rightarrow \quad y = \frac{18 - x}{3}
\]

#### Step 2: Substitute this expression for \( y \) in the other two equations and solve. I'll work through this calculation for you.It seems there is no exact solution to this system of equations, which suggests that a mistake may have occurred when setting up the equations or that there might be a conflict in the nutritional requirements.

Let me verify the setup and approach again, adjusting for any possible issues. Here's a revised approach based on eliminating one variable systematically:

- Simplify the second and third equations to reduce the system further, and attempt a new solution.

I'll recalculate this for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can proceed by manually solving the system of equations using substitution or elimination:

1. Solve one of the simpler equations for one variable (such as solving \( 2x + y = 22 \) for \( y \)).
2. Substitute that expression into the other two equations.
3. Solve for \( x \) and \( y \).

Would you like a detailed explanation on how to solve it manually?

Here are 5 questions that can help expand your understanding:
1. What are substitution and elimination methods in solving systems of equations?
2. How do you check for inconsistent systems of equations?
3. Can systems of equations have no solution or infinitely many solutions?
4. What methods can you use to solve more complex systems involving three or more variables?
5. How do constraints like minimizing weight impact solving real-world problems like this?

**Tip:** Always double-check the consistency of your system before solving to ensure it reflects real-world constraints.

A nutritionist, working for NASA, must meet certain minimum nutritional requirements and yet keep the weight of the food at a minimum. He is considering a combination of two foods, which are packaged in tubes. Each tube of food A contains 4 units of protein, 2 units of carbohydrates, and 2 units of fat and weighs 2 pounds. Each tube of food B contains 3 units of protein, 6 units of carbohydrates, and 1 unit of fat and weighs 2 pounds. The requirement calls for 54 units of protein, 36 units of carbohydrates, and 22 units of fat. How many tubes of each food should be supplied to the astronauts?

This problem focuses on solving a system of linear equations to meet specific nutritional requirements for astronauts. Learn how to calculate the number of tubes of food A and B based on protein, carbohydrate, and fat content. Suitable for high school students in Grades 10-12.

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