To solve the given system of linear equations using the augmented matrix:
[tex]\[
\begin{bmatrix}
1 & 2 & -12 \\
0 & 1 & -5
\end{bmatrix}
\][/tex]
we can perform back substitution since the matrix is already in row-echelon form.
First, let's interpret the augmented matrix as a system of linear equations:
[tex]\[
\begin{cases}
x + 2y = -12, \\
y = -5
\end{cases}
\][/tex]
From the second equation, we have:
[tex]\[ y = -5 \][/tex]
With this value of \( y \), we can substitute it back into the first equation to find \( x \):
[tex]\[
x + 2(-5) = -12 \\
x - 10 = -12 \\
x = -12 + 10 \\
x = -2
\][/tex]
So, the values of \( x \) and \( y \) are:
[tex]\[ x = -2 \][/tex]
[tex]\[ y = -5 \][/tex]
Therefore, the solution set is the ordered pair:
[tex]\[ (-2, -5) \][/tex]
Considering the given options, the correct choice is:
A. There is one solution. The solution set is [tex]\(\{ (-2, -5) \} \)[/tex].
