This problem can be solved using a system of linear equations. Let's define variables based on the number of tubes of each type of food:

• Let $x$ be the number of tubes of food A.
• Let $y$ be the number of tubes of food B.

The nutritional and weight requirements for the astronauts are:

Protein Requirement:

• Each tube of food A provides 4 units of protein.
• Each tube of food B provides 3 units of protein.
• The total required protein is 42 units.

This gives the equation: $4x + 3y = 42$

Carbohydrates Requirement:

• Each tube of food A provides 2 units of carbohydrates.
• Each tube of food B provides 6 units of carbohydrates.
• The total required carbohydrates are 48 units.

This gives the equation: $2x + 6y = 48$

Fat Requirement:

• Each tube of food A provides 2 units of fat.
• Each tube of food B provides 1 unit of fat.
• The total required fat is 16 units.

This gives the equation: $2x + y = 16$

Now we have a system of three equations:

1. $4x + 3y = 42$
2. $2x + 6y = 48$
3. $2x + y = 16$

Solving the system of equations:

Step 1: Solve the third equation for $y$:

From $2x + y = 16$, we can solve for $y$: $y = 16 - 2x$

Step 2: Substitute $y = 16 - 2x$ into the first equation:

Substituting into $4x + 3y = 42$: $4x + 3(16 - 2x) = 42$ Expand and solve for $x$: $4x + 48 - 6x = 42$ $-2x + 48 = 42$ $-2x = -6$ $x = 3$

Step 3: Substitute $x = 3$ into $y = 16 - 2x$:

$y = 16 - 2(3) = 16 - 6 = 10$

Final Answer:

• The number of tubes of food A is $x = 3$.
• The number of tubes of food B is $y = 10$.

Thus, the astronauts should be supplied with 3 tubes of food A and 10 tubes of food B.

Would you like further details or have any questions?

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Here are 5 related questions to expand on this problem:

1. How does the total weight of the food change with different combinations of food A and food B?
2. How would the solution change if the nutritional requirements were increased by 20%?
3. What would happen if food A had a different weight, say 3 pounds per tube?
4. How could you solve this problem graphically using the system of equations?
5. Can this problem be modeled using a linear programming approach to minimize weight?

Tip: When solving systems of equations, substitution and elimination methods are helpful strategies to simplify the problem step-by-step.