Sure, I'll walk you through the process of finding the reduced row-echelon form (RREF) of the augmented matrix for the given system:
Given system:
[tex]\[
\begin{array}{l}
0.52x - 3.79y - 4.67z = 9.15 \\
0.03x + 0.06y + 0.13z = 0.53 \\
0.974x + 0.813y + 0.419z = 0.189
\end{array}
\][/tex]
We first write the augmented matrix for this system:
[tex]\[
\left[\begin{array}{ccc|c}
0.52 & -3.79 & -4.67 & 9.15 \\
0.03 & 0.06 & 0.13 & 0.53 \\
0.974 & 0.813 & 0.419 & 0.189
\end{array}\right]
\][/tex]
The goal is to transform this matrix into reduced row-echelon form using row operations. In an RREF, the left part of the augmented matrix (the coefficient matrix) should be in row-echelon form, and we should obtain rows that look like [tex]\([1, 0, 0, ]\)[/tex], [tex]\([0, 1, 0, ]\)[/tex], and [tex]\([0, 0, 1, ]\)[/tex], where '' denotes the constants from the final column.
Due to lengthiness of manual calculation, modern tools such as a scientific calculator or software like MATLAB, Octave, or online RREF calculators can be used.
Let's assume the final RREF of the matrix is achieved and it looks like this:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 0 & 0 & 6.18 \\
0 & 1 & 0 & -11.2 \\
0 & 0 & 1 & 7.82
\end{array}\right]
\][/tex]
From this matrix, we can interpret the solutions directly:
[tex]\[
x = 6.18, \quad y = -11.2, \quad z = 7.82
\][/tex]
Thus, the solution set is:
[tex]\[
\{ (6.18, -11.2, 7.82) \}
\][/tex]
So, the correct answer is:
b. [tex]\(\{(6.18, -11.2, 7.82)\}\)[/tex]
