To solve the system of equations using matrix methods, we first need to put the system into an augmented matrix form. The system of equations is:
[tex]\[
\begin{cases}
3x - y = 4 & (1) \\
6x - 2y = 7 & (2)
\end{cases}
\][/tex]
The augmented matrix for this system of equations will be:
[tex]\[
\begin{pmatrix}
3 & -1 & | & 4 \\
6 & -2 & | & 7
\end{pmatrix}
\][/tex]
We will use row operations to attempt to find a solution. Let's make the augmented matrix row-echelon form.
First, observe the relationship between the rows. Notice that the second row is just twice the first row when considering the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
\begin{pmatrix}
3 & -1 & | & 4 \\
6 & -2 & | & 7
\end{pmatrix}
\][/tex]
[tex]\[ R2 = 2 \cdot R1 \][/tex]
However, if we double the first row, we get:
[tex]\[
\begin{cases}
6x - 2y = 8
\end{cases}
\][/tex]
Comparing these, we see that:
[tex]\[
\begin{pmatrix}
6 & -2 & | & 8 \\
6 & -2 & | & 7
\end{pmatrix}
\][/tex]
Since [tex]\(8 \neq 7\)[/tex], the equations are inconsistent. This means there is no single pair [tex]\((x, y)\)[/tex] that satisfies both equations simultaneously.
Thus, the system has no solution.
This leads us to the correct answer:
C. No solution
