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Use the Gauss-Jordan method to solve the following system of equations.

[tex]\[
\begin{array}{r}
8x - 3y = 7 \\
16x - 6y = 1
\end{array}
\][/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The solution is [tex]\(\boxed{\text{(ordered pair)}}\)[/tex].

B. There are infinitely many solutions. The solution is [tex]\(\boxed{(x, y)}\)[/tex], where [tex]\(y\)[/tex] is any real number. [tex]\(\boxed{x}\)[/tex].

C. There is no solution.



Answer :

GinnyAnswer
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.

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