To determine the correct matrix representation for the system of equations, let's first establish the system based on the given information. We have three equations with three variables [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex], representing the servings of each type of feed. The equations are derived from the grams of fat, carbohydrates, and protein each type of feed provides:
1. Fat equation:
[tex]\[ 11x + 4y + 10z = 44 \][/tex]
2. Carbohydrate equation:
[tex]\[ 24x + 8y = 48 \][/tex]
3. Protein equation:
[tex]\[ 3x + 31y + 18z = 132 \][/tex]
We want to represent this system in a matrix form, specifically an augmented matrix, which includes the coefficients of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex], as well as the constants on the right side of the equations.
Firstly, let's put down the matrix elements systematically from the equations:
- The coefficients from the fat equation provide the first row.
- The coefficients from the carbohydrate equation give us the second row.
- The coefficients from the protein equation result in the third row.
Thus, the augmented matrix is:
[tex]\[ \left[\begin{array}{ccc|c}
11 & 4 & 10 & 44 \\
24 & 8 & 0 & 48 \\
3 & 31 & 18 & 132
\end{array}\right] \][/tex]
We now compare this matrix to the options provided:
- Option A is exactly:
[tex]\[ \left[\begin{array}{ccc|c}
11 & 4 & 10 & 44 \\
24 & 8 & 0 & 48 \\
3 & 31 & 18 & 132
\end{array}\right] \][/tex]
- Option B is missing the constants on the right side of the augmented matrix, so it cannot be the correct representation:
[tex]\[ \left[\begin{array}{ccc}
11 & 4 & 10 \\
24 & 8 & 0 \\
3 & 31 & 18
\end{array}\right] \][/tex]
- Option C has incorrect values for the constants on the right side:
[tex]\[ \left[\begin{array}{ccc|c}
11 & 4 & 10 & 3 \\
24 & 8 & 0 & 31 \\
44 & 48 & 132 & 18
\end{array}\right] \][/tex]
- Option D has the mixed order of coefficients and thus does not represent the actual system:
[tex]\[ \left[\begin{array}{ccc|c}
11 & 24 & 3 & 44 \\
4 & 8 & 31 & 48 \\
10 & 0 & 18 & 132
\end{array}\right] \][/tex]
Therefore, the correct matrix that represents the system of equations is Option A.
