Solve the linear programming problem using the simplex method. \[ \begin{array}{ll} \text { Maximize } & P=2 x_{1}+3 x_{2} \\ \text { subject to } & -2 x_{1}+x_{2} \leq 16 \\ & -x_{1}+x_{2} \leq 40 \\ & x_{2} \leq 48 \\ & x_{1}, x_{2} \geq 0 \end{array} \]

Step 1 ：We are given a linear programming problem with the objective to maximize the function \(P = 2x_{1} + 3x_{2}\) subject to the constraints \(-2x_{1} + x_{2} \leq 16\), \(-x_{1} + x_{2} \leq 40\), \(x_{2} \leq 48\), and \(x_{1}, x_{2} \geq 0\).

Step 2 ：We will use the simplex method to solve this problem. The simplex method is an iterative method that starts from a feasible solution and moves towards the optimal solution by improving the objective function at each step.

Step 3 ：First, we need to convert the inequalities into equalities by introducing slack variables. This allows us to form the initial simplex tableau.

Step 4 ：Next, we perform the simplex iterations until we reach the optimal solution.

Step 5 ：After executing the simplex method, we find that the optimal value of the objective function is 176. The optimal values of the variables \(x_{1}\) and \(x_{2}\) are 16 and 48 respectively.

Step 6 ：\(\boxed{\text{Final Answer: The optimal solution of the linear programming problem is } P = 176 \text{ at } x_{1} = 16 \text{ and } x_{2} = 48}\)