A region is given by x + 2y ≤ 10, 3x + y ≤ 15, x ≥ 0 and y ≥ 0…

A region is given by $x + 2y \le 10$, $3x + y \le 15$, $x \ge 0$ and $y \ge 0$. Its corners are $(0, 0)$, $(5, 0)$, $(4, 3)$ and $(0, 5)$. The objective function is $Z = x + ay$, where $a$ is a positive number. Find the value of $a$ for which $Z$ is the same at $(4, 3)$ and at $(0, 5)$. For that $a$, find the maximum of $Z$ over the region and the number of corners where it occurs.