Let's solve the given system of equations using the Gauss-Jordan elimination method:
[tex]\[
\begin{cases}
8x - 3y = 7 \\
16x - 6y = 1
\end{cases}
\][/tex]
First, we write the augmented matrix for the system:
[tex]\[
\left[\begin{array}{ccc}
8 & -3 & | & 7 \\
16 & -6 & | & 1
\end{array}\right]
\][/tex]
### Step 1: Normalize the first row
We divide the first row by 8 to make the leading coefficient 1.
[tex]\[
\left[\begin{array}{ccc}
1 & -\frac{3}{8} & | & \frac{7}{8} \\
16 & -6 & | & 1
\end{array}\right]
\][/tex]
### Step 2: Eliminate the first column of the second row
Next, we subtract 16 times the first row from the second row to eliminate the x-term in the second equation.
[tex]\[
R_{2} = R_{2} - 16R_{1}
\][/tex]
[tex]\[
\left[\begin{array}{ccc}
1 & -\frac{3}{8} & | & \frac{7}{8} \\
0 & 0 & | & -\frac{25}{8}
\end{array}\right]
\][/tex]
### Analysis of the resulting matrix
The second row translates into the equation:
[tex]\[
0 = -\frac{25}{8}
\][/tex]
This is a contradiction because [tex]\( 0 \)[/tex] cannot be equal to [tex]\( -\frac{25}{8} \)[/tex].
### Conclusion
Since we have arrived at a contradiction, this system of equations has no solutions.
The correct choice is:
C. There is no solution.
