Solve the following system of equations using Gaussian or Gauss-Jordan elimination. \( \begin{array}{l}-25 x-3 y=-113 \\ 10 x+y=46\end{array} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There is only solution. The solution to the system of equations is (Simplify your answer. Type an ardered pair) B. There are irfnitely mary solufions of the form ( \( x \). (Type an expression using \( x \) as the vaniable) C. There is no solution.

To solve the system of equations using Gaussian elimination, let's set up the augmented matrix: \[ \begin{pmatrix} -25 & -3 & | & -113 \\ 10 & 1 & | & 46 \end{pmatrix} \] Now, we will perform row operations to reach row echelon form. We'll multiply the first row by \( -\frac{1}{25} \) to simplify it: \[ \begin{pmatrix} 1 & \frac{3}{25} & | & \frac{113}{25} \\ 10 & 1 & | & 46 \end{pmatrix} \] Next, we want to eliminate \( 10 \) in the second row's first column. We can do this by replacing the second row with \( R_2 - 10R_1 \): \[ R_2 \rightarrow R_2 - 10R_1 \] So we calculate it as follows: \[ \begin{pmatrix} 1 & \frac{3}{25} & | & \frac{113}{25} \\ 0 & 1 - 10\left(\frac{3}{25}\right) & | & 46 - 10\left(\frac{113}{25}\right) \end{pmatrix} \] Calculating the second row gives us: \[ 1 - \frac{30}{25} = 1 - 1.2 = -0.2 = -\frac{1}{5} \] And: \[ 46 - \frac{1130}{25} = \frac{1150 - 1130}{25} = \frac{20}{25} = \frac{4}{5} \] Now the augmented matrix looks like: \[ \begin{pmatrix} 1 & \frac{3}{25} & | & \frac{113}{25} \\ 0 & -\frac{1}{5} & | & \frac{4}{5} \end{pmatrix} \] To eliminate the fractions, we can multiply the second row by \(-5\): \[ \begin{pmatrix} 1 & \frac{3}{25} & | & \frac{113}{25} \\ 0 & 1 & | & -4 \end{pmatrix} \] Now, we can substitute back. Let's express \( y \): From Row 2: \[ y = -4 \] Substituting \( y = -4 \) back into the first equation from the original system: \[ 10x - 4 = 46 \implies 10x = 46 + 4 \implies 10x = 50 \implies x = 5 \] Thus the solution to the system is: The solution to the system of equations is \( (5, -4) \). A. There is only one solution. The solution to the system of equations is (5, -4).