Solve the following system of linear inequalities using the simplex method and find the maximum value of the objective function: $$\{\begin{array}{l}x+y\le 5\\ 2x+y\le 8\\ x,y\ge 0\end{array}$$ where the objective function is $Z=2x+3y$.

Step 1 ：Step 1: Convert the inequalities to equations by introducing slack variables: $$\{\begin{array}{l}x+y+s_1=5\\ 2x+y+s_2=8\\ x,y,s_1,s_2\ge 0\end{array}$$

Step 2 ：Step 2: Set up the initial simplex tableau: $$\begin{array}{ccccc}1& 1& 1& 0& 5\\ 2& 1& 0& 1& 8\\ -2& -3& 0& 0& 0\end{array}$$

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: $$\begin{array}{ccccc}0& 1& 1& -0.5& 1\\ 1& 0.5& 0& 0.5& 4\\ 0& -1& 0& 1& 8\end{array}$$

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: $$\begin{array}{ccccc}0& 1& 1& -0.5& 1\\ 1& 0& -2& 1& 2\\ 0& 0& -2& 0.5& 9\end{array}$$

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