An animal breeder can buy four types of food for Vietnamese​ pot-bellied pigs. Each case of Brand A contains 2525 units of​ fiber, 3030 units of​ protein, and 3030 units of fat. Each case of Brand B contains 100100 units of​ fiber, 9090 units of​ protein, and 7070 units of fat. Each case of Brand C contains 175175 units of​ fiber, 150150 units of​ protein, and 130130 units of fat. Each case of Brand D contains 300300 units of​ fiber, 270270 units of​ protein, and 190190 units of fat. How many cases of each should the breeder mix to obtain a food that provides 29002900 units of​ fiber, 25502550 units of​ protein, and 20902090 units of​ fat?

We can set up this problem as a system of linear equations to solve for the number of cases of each brand (A, B, C, D) that the breeder should buy.

Let:
- \( x_1 \) be the number of cases of Brand A,
- \( x_2 \) be the number of cases of Brand B,
- \( x_3 \) be the number of cases of Brand C,
- \( x_4 \) be the number of cases of Brand D.

From the given information, we can form the following system of equations based on the fiber, protein, and fat content:

### 1. Fiber equation:
\[
2525x_1 + 100100x_2 + 175175x_3 + 300300x_4 = 29002900
\]

### 2. Protein equation:
\[
3030x_1 + 9090x_2 + 150150x_3 + 270270x_4 = 25502550
\]

### 3. Fat equation:
\[
3030x_1 + 7070x_2 + 130130x_3 + 190190x_4 = 20902090
\]

Now, we can solve this system of equations using substitution, elimination, or matrix methods. I will use matrix methods to solve it for \(x_1, x_2, x_3, x_4\). Let's compute it!The solution for the number of cases of each brand (\(x_1\), \(x_2\), \(x_3\), and \(x_4\)) depends on the number of cases of Brand D (\(x_4\)). The general solution for the system of equations is:

\[
x_1 = \frac{39538499 \cdot x_4}{353399} - \frac{1080108000}{353399}
\]
\[
x_2 = \frac{5005 \cdot x_4}{3499} - \frac{150015}{3499}
\]
\[
x_3 = \frac{820001992}{3502499} - \frac{14501 \cdot x_4}{3499}
\]

Thus, the number of cases for Brands A, B, and C can be expressed in terms of \(x_4\). To get specific values, we would need to assign a value to \(x_4\) (the number of cases of Brand D).

Would you like to assign a specific value to \(x_4\), or have a detailed explanation on how to interpret these equations?

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Here are 5 related questions:
1. How do we interpret the number of cases based on the equations?
2. Can we choose any value for \(x_4\), or are there restrictions?
3. What happens if we set \(x_4 = 0\)? How does that affect \(x_1\), \(x_2\), and \(x_3\)?
4. How would you solve the system using another method, like substitution?
5. How can we check if the solution satisfies the original problem?

**Tip**: In systems with free variables, assigning values to one variable simplifies solving for others.

Learn how to solve a system of equations to determine how many cases of each brand of food a breeder should mix for Vietnamese pot-bellied pigs. This problem uses linear algebra concepts and matrix methods. Suitable for students in Grades 10-12.

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