Solve the following linear programming problem. \[ \begin{array}{ll} \text { Maximize: } & z=4 x+5 y \\ \text { subject to: } & 5 x+4 y \leq 20 \\ & 8 x+y \leq 20 \\ & x \geq 0, y \geq 0 \end{array} \]

Step 1 ：We are given a linear programming problem where we are asked to maximize the function \(z = 4x + 5y\) subject to the constraints \(5x + 4y ≤ 20\), \(8x + y ≤ 20\), and \(x, y ≥ 0\).

Step 2 ：To solve this problem, we can use a method that minimizes the objective function, so to maximize \(z\), we need to minimize \(-z\).

Step 3 ：The optimal value of the objective function \(z\) is 25, achieved when \(x = 0\) and \(y = 5\). These values satisfy all the given constraints.

Step 4 ：Final Answer: The maximum value of the function \(z = 4x + 5y\) subject to the constraints \(5x + 4y ≤ 20\), \(8x + y ≤ 20\), and \(x, y ≥ 0\) is \(z = \boxed{25}\) at the point \((x, y) = (0, 5)\). So, the final answer is \(z = \boxed{25}\), \(x = \boxed{0}\), and \(y = \boxed{5}\).