Step 1 :Let's denote the number of tubes of food A as x and the number of tubes of food B as y. We can then set up the following equations based on the information given in the problem:
Step 2 :\(4x + 3y = 48\) (This equation represents the protein requirement)
Step 3 :\(2x + 6y = 42\) (This equation represents the carbohydrates requirement)
Step 4 :\(2x + y = 18\) (This equation represents the fat requirement)
Step 5 :The system of equations is overdetermined, meaning there are more equations than unknowns. This can sometimes lead to no solution. However, in this case, we can observe that the second equation (carbohydrates) is a multiple of the third equation (fat). This means we can eliminate one of these equations without losing any information.
Step 6 :Let's eliminate the second equation and solve the system again with the remaining two equations:
Step 7 :\(4x + 3y = 48\)
Step 8 :\(2x + y = 18\)
Step 9 :Solving this system of equations gives us the solution: \(x = 12, y = 3\)
Step 10 :Final Answer: The number of tubes of food A is \(\boxed{12}\) and the number of tubes of food B is \(\boxed{3}\)