$\begin{array}{ll}\text { Maximize } & P=7 x_{1}+2 x_{2}-x_{3} \\ \text { subject to } & x_{1}+x_{2}-x_{3} \leq 7 \\ & 2 x_{1}+4 x_{2}+3 x_{3} \leq 21 \\ & x_{1}, x_{2}, x_{3} \geq 0\end{array}$

Step 1 ：Given the linear programming problem: Maximize \(P=7 x_{1}+2 x_{2}-x_{3}\) subject to \(x_{1}+x_{2}-x_{3} \leq 7\), \(2 x_{1}+4 x_{2}+3 x_{3} \leq 21\), and \(x_{1}, x_{2}, x_{3} \geq 0\).

Step 2 ：First, we convert the maximization problem to a minimization problem by multiplying the objective function by -1. So, the objective function becomes \(-P=-7 x_{1}-2 x_{2}+x_{3}\).

Step 3 ：Then, we define the coefficients of the objective function and the inequality constraints. The coefficients of the objective function are [-7, -2, 1]. The coefficients of the inequality constraints are [[1, 1, -1], [2, 4, 3]] and the right-hand side values of the inequality constraints are [7, 21].

Step 4 ：We also define the bounds for the variables \(x_{1}\), \(x_{2}\), and \(x_{3}\). All variables are non-negative, so the lower bound is 0 and the upper bound is infinity.

Step 5 ：Next, we solve the problem using a linear programming solver. The solver returns the optimal value of the objective function and the values of the variables \(x_{1}\), \(x_{2}\), and \(x_{3}\) that achieve this optimal value.

Step 6 ：The optimal value of the objective function is -57.4. Since we multiplied the objective function by -1, the optimal value for the original problem is \(-(-57.4) = 57.4\).

Step 7 ：The values of the variables \(x_{1}\), \(x_{2}\), and \(x_{3}\) that achieve this optimal value are 8.4, 0, and 1.4 respectively.

Step 8 ：Final Answer: The maximum value of the objective function \(P=7 x_{1}+2 x_{2}-x_{3}\) subject to the given constraints is \(\boxed{57.4}\) and the values of \(x_{1}\), \(x_{2}\), and \(x_{3}\) that achieve this maximum value are \(\boxed{8.4}\), \(\boxed{0}\), and \(\boxed{1.4}\) respectively.