To solve the problem of finding the maximum value of the function $z = 5x + 3y$ under the given constraints, we can apply the Linear Programming method.

The constraints provided are:

1. $x \geq 2$
2. $y \geq 3$
3. $4x + 7y \leq 57$

Objective function:

Maximize $z = 5x + 3y$.

Step-by-step approach:

1. Find the feasible region:

   • The region is bounded by the intersection of the constraints, which represent lines or inequalities.
   • We will solve for the points of intersection of the lines formed by the constraints.

2. Check the vertices of the feasible region: The maximum or minimum of the linear function under linear constraints will occur at one of the vertices (corner points) of the feasible region.

We will now calculate the vertices and evaluate the objective function $z = 5x + 3y$ at each vertex.

1. Intersection of $x = 2$ and $4x + 7y = 57$:

Substitute $x = 2$ into $4x + 7y = 57$: $4(2) + 7y = 57 \implies 8 + 7y = 57 \implies 7y = 49 \implies y = 7$ Vertex: $(2, 7)$.

2. Intersection of $y = 3$ and $4x + 7y = 57$:

Substitute $y = 3$ into $4x + 7y = 57$: $4x + 7(3) = 57 \implies 4x + 21 = 57 \implies 4x = 36 \implies x = 9$ Vertex: $(9, 3)$.

3. Intersection of $x = 2$ and $y = 3$:

This is a simple intersection: $x = 2, \quad y = 3$ Vertex: $(2, 3)$.

Now, let's evaluate the objective function at each of these vertices:

• At $(2, 7)$: $z = 5(2) + 3(7) = 10 + 21 = 31$
• At $(9, 3)$: $z = 5(9) + 3(3) = 45 + 9 = 54$
• At $(2, 3)$: $z = 5(2) + 3(3) = 10 + 9 = 19$

Conclusion:

The maximum value of $z = 54$, which occurs at the point $(9, 3)$.

Would you like further details or clarifications?

Related Questions:

1. How can we graph the feasible region for these inequalities?
2. What if the constraints were different—how would the solution method change?
3. Can we use another optimization method like the Simplex method for this problem?
4. How would the solution change if we were minimizing instead of maximizing the objective function?
5. Can constraints like $x \leq 0$ or $y \leq 0$ create unbounded solutions?

Tip:

In linear programming, always check the corner points of the feasible region as the optimal solution lies at one of these points.