Questions: An animal breeder can buy four types of food for Vietnamese pot-bellied pigs. Each case of Brand A contains 35 units of fiber, 40 units of protein, and 40 units of fat. Each case of Brand B contains 70 units of fiber, 40 units of protein, and 30 units of fat. Each case of Brand C contains 105 units of fiber, 40 units of protein, and 30 units of fat. Each case of Brand D contains 210 units of fiber, 120 units of protein, and 80 units of fat. How many cases of each should the breeder mix to obtain a food that provides 2100 units of fiber, 1000 units of protein, and 750 units of fat?

Solution Steps

To solve this problem, we need to determine which combination of cases of food brands A, B, C, and D satisfies the given nutritional requirements. This involves solving a system of linear equations for each option provided. We will substitute each option into the equations to check if they satisfy the conditions.

We are given a system of linear equations representing the nutritional content of different brands of pig food. The equations are:

\[ \begin{align_} 35x + 70y + 105z + 210w &= 2100 \\ 40x + 40y + 40z + 120w &= 1000 \\ 40x + 30y + 30z + 80w &= 750 \end{align_} \]

where \(x\), \(y\), \(z\), and \(w\) represent the number of cases of brands A, B, C, and D, respectively.

We need to evaluate each given option to see if it satisfies the system of equations. The options are:

• A: \(x = 2\), \(y = 5\), \(z = 2\), \(w = 12\)
• B: \(x = 0\), \(y = 15\), \(z = 10\), \(w = 0\)
• C: \(x = 5\), \(y = 15\), \(z = 5\), \(w = 0\)
• D: \(x = 5\), \(y = 0\), \(z = 12\), \(w = 0\)
• E: \(x = 1\), \(y = 10\), \(z = 11\), \(w = 1\)
• F: \(x = 2\), \(y = 5\), \(z = 12\), \(w = 2\)
• G: \(x = 0\), \(y = 0\), \(z = 10\), \(w = 15\)
• H: \(x = 3\), \(y = 0\), \(z = 13\), \(w = 3\)

Substitute each option into the system of equations to check if they satisfy all three equations. The valid options are those that satisfy all equations simultaneously.

After evaluating each option, the valid options that satisfy the system of equations are:

• B: \(x = 0\), \(y = 15\), \(z = 10\), \(w = 0\)
• E: \(x = 1\), \(y = 10\), \(z = 11\), \(w = 1\)
• F: \(x = 2\), \(y = 5\), \(z = 12\), \(w = 2\)
• H: \(x = 3\), \(y = 0\), \(z = 13\), \(w = 3\)

Final Answer