To solve this problem, we need to determine how many cups of rice, tofu, and peanuts Bill should eat to meet the following nutritional goals:

• Carbohydrates: 150 grams
• Fat: 262 grams
• Protein: 169 grams

Given the nutritional content per cup for each food item:

1. Rice provides:

   • Carbohydrates: 44 g
   • Fat: 0 g
   • Protein: 1 g

2. Tofu provides:

   • Carbohydrates: 4 g
   • Fat: 13 g
   • Protein: 18 g

3. Peanuts provide:

   • Carbohydrates: 30 g
   • Fat: 70 g
   • Protein: 32 g

Let $x$ be the number of cups of rice, $y$ the number of cups of tofu, and $z$ the number of cups of peanuts. We now set up a system of equations based on the nutritional goals:

1. Carbohydrates Equation:
   $44x + 4y + 30z = 150$

2. Fat Equation:
   $0x + 13y + 70z = 262$

3. Protein Equation:
   $1x + 18y + 32z = 169$

We need to solve this system of equations.

Let's solve it step-by-step.

1. Carbohydrates Equation: $44x + 4y + 30z = 150$
2. Fat Equation: $13y + 70z = 262$
3. Protein Equation: $x + 18y + 32z = 169$

I'll solve this system of equations now. sympy import symbols, Eq, solve

Define variables

x, y, z = symbols('x y z')

Define the equations

eq1 = Eq(44x + 4y + 30z, 150) # Carbohydrates equation eq2 = Eq(13y + 70z, 262) # Fat equation eq3 = Eq(x + 18y + 32*z, 169) # Protein equation

Solve the system of equations

solution = solve((eq1, eq2, eq3), (x, y, z)) solutionThe solution to the system of equations is:

• Cups of rice ($x$) = 1
• Cups of tofu ($y$) = 4
• Cups of peanuts ($z$) = 3

Thus, Bill should eat 1 cup of rice, 4 cups of tofu, and 3 cups of peanuts to meet his nutritional goals.

Does this answer your question? Let me know if you need further clarification or a step-by-step explanation!

Related Questions:

1. How can we verify the solution by substituting it back into the equations?
2. What would the solution be if Bill's nutritional goals changed to 200 g carbs, 300 g fat, and 180 g protein?
3. How do we graphically represent the solution to this system?
4. How can we use matrices to solve this system of equations?
5. What happens if there are no solutions or multiple solutions for a similar system?

Tip:

When solving systems of equations, double-check by substituting the solution back into the original equations to ensure they hold true.