Maximize $p=x+2 y$ subject to \[ \begin{array}{c} x+4 y \leq 9 \\ 5 x+y \leq 7 \\ x \geq 0, y \geq 0 \\ p=(x) \\ (x, y)=\left(\begin{array}{l} x \\ x \end{array}\right) \end{array} \]

Step 1 ：Given the problem to maximize \(p=x+2y\) subject to the constraints \(x+4y\leq9\), \(5x+y\leq7\), and \(x\geq0\), \(y\geq0\).

Step 2 ：We can solve this problem using linear programming.

Step 3 ：First, we define the coefficients of the objective function as \(c = [-1, -2]\).

Step 4 ：Next, we define the coefficients of the inequality constraints as \(A_{ub} = [[1, 4], [5, 1]]\) and the right-hand side of the inequality constraints as \(b_{ub} = [9, 7]\).

Step 5 ：We also set the bounds for the variables \(x\) and \(y\) as non-negative, i.e., \(x, y \geq 0\).

Step 6 ：By solving the linear programming problem, we find the optimal values are \(x=1\), \(y=2\), and \(p=5\).

Step 7 ：These values satisfy all the constraints and maximize the objective function \(p=x+2y\).

Step 8 ：Final Answer: The maximum value of \(p=x+2y\) subject to the constraints \(x+4y\leq9\), \(5x+y\leq7\), and \(x\geq0\), \(y\geq0\) is \(p=5\) at the point \((x, y)=(1, 2)\). So, the final answer is \(\boxed{5}\).