Solve the following system of linear inequalities using the simplex method and find the maximum value of the objective function: \[ \begin{cases} x + y \leq 5 \\ 2x + y \leq 8 \\ x, y \geq 0 \end{cases} \] where the objective function is \( Z = 2x + 3y \).

Step 1 ：Step 1: Convert the inequalities to equations by introducing slack variables: \[ \begin{cases} x + y + s_1 = 5 \\ 2x + y + s_2 = 8 \\ x, y, s_1, s_2 \geq 0 \end{cases} \]

Step 2 ：Step 2: Set up the initial simplex tableau: \[ \begin{array}{cccc|c} 1 & 1 & 1 & 0 & 5 \\ 2 & 1 & 0 & 1 & 8 \\ \hline -2 & -3 & 0 & 0 & 0 \end{array} \]

Step 3 ：Step 3: Perform the pivot operation. The pivot element is the one in the first column and second row. After the pivot operation, the new simplex tableau is: \[ \begin{array}{cccc|c} 0 & 1 & 1 & -0.5 & 1 \\ 1 & 0.5 & 0 & 0.5 & 4 \\ \hline 0 & -1 & 0 & 1 & 8 \end{array} \]

Step 4 ：Step 4: Perform the pivot operation again. The pivot element is the one in the second column and first row. After the pivot operation, the final simplex tableau is: \[ \begin{array}{cccc|c} 0 & 1 & 1 & -0.5 & 1 \\ 1 & 0 & -2 & 1 & 2 \\ \hline 0 & 0 & -2 & 0.5 & 9 \end{array} \]

Step 5 ：Step 5: The final solution is obtained by reading the last column of the tableau: \( x = 2, y = 1 \).