The linear programming problem provided is as follows:

Objective Function: Maximize $Z = 2x + 5y$

Subject to the following constraints:

1. $2x + y \leq 6$
2. $x + y \leq 5$
3. $x + 3y \geq 3$
4. $x \geq 0$
5. $y \geq 0$

Now, let's solve this step by step. We can use graphical methods or algebraic methods to solve the problem. To begin with, we will start by plotting the constraints to find the feasible region and then evaluate the objective function at the vertices of the feasible region.

Would you like me to walk you through solving this using a graphical approach, or would you prefer a more algebraic method, like the Simplex method?