Solve the linear programming problem using the simplex method \[ \begin{array}{ll} \text { Maximize } & P=6 x_{1}+2 x_{2}-x_{3} \\ \text { subject to } & x_{1}+x_{2}-x_{3} \leq 2 \\ & 2 x_{1}+4 x_{2}+3 x_{3} \leq 6 \\ & x_{1}, x_{2}, x_{3} \geq 0 \end{array} \]

Step 1 ：Given the linear programming problem: Maximize \(P=6 x_{1}+2 x_{2}-x_{3}\) subject to \(x_{1}+x_{2}-x_{3} \leq 2\), \(2 x_{1}+4 x_{2}+3 x_{3} \leq 6\), and \(x_{1}, x_{2}, x_{3} \geq 0\).

Step 2 ：We 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 ：By applying the simplex method, we find that the optimal value of the objective function is 14, and the values of \(x_{1}\), \(x_{2}\), and \(x_{3}\) at the optimal solution are 2.4, 0, and 0.4 respectively.

Step 4 ：\(\boxed{\text{Final Answer: The optimal value of the objective function is 14, and the values of } x_{1}, x_{2}, \text{ and } x_{3} \text{ at the optimal solution are 2.4, 0, and 0.4 respectively.}}\)