Step-by-step explanation:

To find the minimum value of the objective function 3x + 5y subject to the given constraints, you can use linear programming techniques. In this case, you have the following constraints:

1. x + 3y ≥ 3

2. x + y ≥ 2

3. x ≥ 0

4. y ≥ 0

First, graph the feasible region defined by these constraints on a coordinate plane.

Start with the constraint x + 3y ≥ 3:

x + 3y = 3

This line passes through points (3, 0) and (0, 1).

Next, consider the constraint x + y ≥ 2:

x + y = 2

This line passes through points (2, 0) and (0, 2).

Now, plot these lines on the coordinate plane and shade the region that satisfies all constraints.

The feasible region is the shaded region in the graph.

Next, you need to evaluate the objective function 3x + 5y at the vertices of this feasible region to find the minimum value.

1. At the point (2, 0):

3x + 5y = 3(2) + 5(0) = 6

2. At the point (0, 1):

3x + 5y = 3(0) + 5(1) = 5

3. At the point (3, 0):

3x + 5y = 3(3) + 5(0) = 9

Now, compare the values of the objective function at these vertices. The minimum value is 5, which occurs at the point (0, 1).

So, the minimum value of 3x + 5y subject to the given constraints is 5, and it is achieved when x = 0 and y = 1