To solve the given system of equations, we need to bring the augmented matrix to its reduced row-echelon form (RREF). The original system of equations is represented by the matrix:
[tex]\[
\left[\begin{array}{ccc|c}
2 & 1 & 1 & 4 \\
-3 & 2 & -1 & -8 \\
1 & -1 & 1 & 5
\end{array}\right]
\][/tex]
Following the step-by-step process of Gaussian elimination, the solution to the system can be represented as:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 0 & 0 & 1 \\
0 & 1 & 0 & -1 \\
0 & 0 & 1 & 3
\end{array}\right]
\][/tex]
This tells us that:
- The first variable [tex]\( x_1 \)[/tex] is equal to 1,
- The second variable [tex]\( x_2 \)[/tex] is equal to -1,
- The third variable [tex]\( x_3 \)[/tex] is equal to 3.
Therefore, the correct matrix representing the solution to this system of equations is:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 0 & 0 & 1 \\
0 & 1 & 0 & -1 \\
0 & 0 & 1 & 3
\end{array}\right]
\][/tex]
The other given matrix:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & -4 \\
0 & 0 & 1 & 3
\end{array}\right]
\][/tex]
does not represent the solutions to the system, as the values in the last column (the constants) do not match the solution we derived.
