The maximum value of the function $z = 7x + 5y$, subject to the constraints $x \le 3, y \le 2, x \ge 0, y \ge 0$ is :

Step 1: Identify the objective function and constraints

The problem asks us to find the maximum value of the objective function $z = 7x + 5y$. This is a linear programming problem, and we need to find the feasible region defined by the given constraints. The constraints are $x \le 3$, $y \le 2$, $x \ge 0$, and $y \ge 0$.

Step 2: Graph the feasible region

The constraints $x \ge 0$ and $y \ge 0$ restrict the feasible region to the first quadrant. The constraint $x \le 3$ means the region is to the left of or on the line $x=3$. The constraint $y \le 2$ means the region is below or on the line $y=2$. Combining these, the feasible region is a rectangle with vertices at $(0{,}0)$, $(3{,}0)$, $(3{,}2)$, and $(0{,}2)$.

Step 3: Evaluate the objective function at each vertex

According to the corner point theorem in linear programming, the maximum or minimum value of the objective function, if it exists, will occur at one of the vertices of the feasible region. We evaluate the objective function $z = 7x + 5y$ at each of the four vertices.

Step 4: Determine the maximum value

Comparing the values of $z$ at the vertices, we find that the maximum value is $31$, which occurs at the point $(3{,}2)$.

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