To convert the given augmented matrix to a linear system, let's first consider the provided matrix:
[tex]\[
\left[\begin{array}{cc|c}
6 & -12 & 21 \\
-9 & 3 & -3
\end{array}\right]
\][/tex]
This augmented matrix represents two linear equations. Each row corresponds to one equation in the system, and the columns represent the coefficients of the variables followed by the constants on the right side of the equations.
### Step-by-Step Solution:
1. First Row Interpretation:
The first row of the matrix is:
[tex]\[
[6, -12, 21]
\][/tex]
This indicates the linear equation:
[tex]\[
6x - 12y = 21
\][/tex]
2. Second Row Interpretation:
The second row of the matrix is:
[tex]\[
[-9, 3, -3]
\][/tex]
This indicates the linear equation:
[tex]\[
-9x + 3y = -3
\][/tex]
### Resultant Linear System:
Combining these interpretations, the resultant linear system of equations is:
[tex]\[
\begin{cases}
6x - 12y = 21 \\
-9x + 3y = -3
\end{cases}
\][/tex]
These are the linear equations derived from the given augmented matrix.
