Solve the following system of equations using matrix elimination method: \(3x - 2y = 7\) and \(5x + y = 11\)

Step 1 ：First, we write this system of equations in matrix form: \(\begin{bmatrix} 3 & -2\ 5 & 1\end{bmatrix} \begin{bmatrix} x\ y\end{bmatrix} = \begin{bmatrix} 7\ 11\end{bmatrix}\)

Step 2 ：Next, we perform row operations to transform the coefficient matrix into row-echelon form. Multiply the first row by 5 and the second row by 3: \(\begin{bmatrix} 15 & -10\ 15 & 3\end{bmatrix} \begin{bmatrix} x\ y\end{bmatrix} = \begin{bmatrix} 35\ 33\end{bmatrix}\)

Step 3 ：Subtract the second row from the first row: \(\begin{bmatrix} 0 & -13\ 15 & 3\end{bmatrix} \begin{bmatrix} x\ y\end{bmatrix} = \begin{bmatrix} 2\ 33\end{bmatrix}\)

Step 4 ：Divide the first row by -13 and the second row by 15 to get the identity matrix: \(\begin{bmatrix} 0 & 1\ 1 & 0\end{bmatrix} \begin{bmatrix} x\ y\end{bmatrix} = \begin{bmatrix} -\frac{2}{13}\ \frac{33}{15}\end{bmatrix}\)

Step 5 ：Finally, we can see that x = \(\frac{33}{15}\) and y = -\(\frac{2}{13}\)