$\begin{array}{cc}\text { Maximize } & P=30 x_{1}+40 x_{2} \\ \text { subject to } & 2 x_{1}+x_{2} \leq 10 \\ & x_{1}+x_{2} \leq 7 \\ & x_{1}+2 x_{2} \leq 12 \\ & x_{1}, x_{2} \geq 0\end{array}$
Solution
Step 1 ：Given a linear programming problem, we are asked to maximize the objective function \(P=30x_1+40x_2\) subject to the constraints \(2x_1+x_2 \leq 10\), \(x_1+x_2 \leq 7\), \(x_1+2x_2 \leq 12\), and \(x_1, x_2 \geq 0\).
Step 2 ：We convert the problem to the standard form for linear programming problems, which is minimize \(c^T x\) subject to \(Ax \leq b\) and \(x \geq 0\). Here, \(c\) is the coefficients of the objective function, \(A\) is the matrix of coefficients of the constraints, \(b\) is the right-hand side of the constraints, and \(x\) is the vector of variables.
Step 3 ：In this case, \(c = [-30, -40]\) (we negate the coefficients because we want to maximize the objective function), \(A = [[2, 1], [1, 1], [1, 2]]\), and \(b = [10, 7, 12]\). The constraints \(x_1, x_2 \geq 0\) are automatically enforced in the linear programming problem.
Step 4 ：After solving the problem, we find that the optimal solution is \(x_1 = 2\), \(x_2 = 5\), and the maximum value of \(P\) is \(260\).
Step 5 ：So, the final answer is \(\boxed{x_1 = 2}\), \(\boxed{x_2 = 5}\), and \(\boxed{P = 260}\).
