The Constraints Of A Problem Are Listed Below. What Are The Vertices Of The Feasible Region? 2x + 3y ≥ 12, 5x + 2y ≥ 15, X ≥ 0, Y ≥ 0

Jun 19, 2025 by ADMIN 134 views

The Constraints of a Problem What Are the Vertices of the Feasible Region?

In the realm of mathematical optimization, understanding constraints and feasible regions is paramount. Constraints define the limitations within which a solution must exist, while the feasible region represents the set of all possible solutions that satisfy these constraints. Within the feasible region, specific points known as vertices play a crucial role in identifying optimal solutions. This article delves into the process of determining the vertices of a feasible region defined by a system of linear inequalities. We will explore the graphical method for visualizing the feasible region and the algebraic techniques for calculating the coordinates of the vertices. By understanding these concepts, you will gain a solid foundation for solving linear programming problems and making informed decisions in various fields, such as resource allocation, production planning, and financial modeling. Linear programming is a powerful tool for optimization, allowing us to find the best possible solution within a given set of constraints. This process often involves maximizing or minimizing an objective function, such as profit or cost, subject to a set of linear inequalities that represent limitations on resources or requirements. The vertices of the feasible region are critical because the optimal solution, if it exists, will always occur at one of these vertices. Therefore, accurately identifying and calculating the vertices is a crucial step in solving linear programming problems. This article aims to provide a clear and comprehensive guide to this process, ensuring that you can confidently tackle a wide range of optimization challenges.

Problem Statement

Let's consider the following system of linear inequalities:

2x + 3y ≥ 12
5x + 2y ≥ 15
x ≥ 0
y ≥ 0

The objective is to find the vertices of the feasible region defined by these constraints. The constraints represent boundaries within the x-y plane, and the feasible region is the area where all constraints are simultaneously satisfied. The vertices are the points where these boundary lines intersect, forming the corners of the feasible region. These points are crucial because they represent the potential locations of optimal solutions for a linear programming problem. The non-negativity constraints, x ≥ 0 and y ≥ 0, restrict the feasible region to the first quadrant of the coordinate plane, which simplifies the analysis. However, even within this quadrant, the feasible region can take various shapes, depending on the other constraints. Understanding how to determine the vertices is essential for identifying the extreme points of this region, which are critical for optimization. This article will guide you through the process of finding these vertices step by step, ensuring that you can apply the same techniques to other similar problems.

Graphical Representation

To visualize the feasible region, we can plot the lines corresponding to the inequalities on the x-y plane.

2x + 3y = 12
5x + 2y = 15
x = 0
y = 0

By plotting these lines, we can identify the region that satisfies all the inequalities. The feasible region is the area bounded by these lines, where the solution exists. Each inequality represents a half-plane, and the feasible region is the intersection of these half-planes. The lines 2x + 3y = 12 and 5x + 2y = 15 are particularly important as they define the main boundaries of the feasible region. The lines x = 0 and y = 0 represent the y-axis and x-axis, respectively, and they restrict the feasible region to the first quadrant. The graphical representation provides a visual understanding of the feasible region, making it easier to identify the vertices. The vertices are the points where these lines intersect, and they are the key to finding the optimal solutions for the problem. By understanding the graphical representation, you can quickly identify the potential vertices and proceed with the algebraic calculations to determine their exact coordinates.

Identifying the Vertices

The vertices of the feasible region are the points where the constraint lines intersect. From the graph, we can identify the following potential vertices:

Intersection of 2x + 3y = 12 and 5x + 2y = 15
Intersection of 2x + 3y = 12 and y = 0
Intersection of 5x + 2y = 15 and x = 0

These points represent the corners of the feasible region and are critical for finding the optimal solution to the problem. Each vertex corresponds to the solution of a system of two linear equations, which can be solved using algebraic techniques such as substitution or elimination. The intersection of 2x + 3y = 12 and 5x + 2y = 15 represents the point where both constraints are simultaneously satisfied, and it is a key vertex for the feasible region. The intersections with the x and y axes, represented by the lines y = 0 and x = 0, respectively, are also important as they define the boundaries of the region within the first quadrant. Identifying these vertices is a crucial step in solving linear programming problems, as the optimal solution will always occur at one of these points. This section provides a clear understanding of how to identify the potential vertices based on the graphical representation of the constraints.

Calculating the Vertices

Now, let's calculate the coordinates of these vertices.

Intersection of 2x + 3y = 12 and 5x + 2y = 15

To find the intersection, we can solve the system of equations:
```
2x + 3y = 12
5x + 2y = 15
```
Multiply the first equation by 5 and the second equation by 2 to eliminate x:
```
10x + 15y = 60
10x + 4y = 30
```
Subtract the second equation from the first:
```
11y = 30
y = 30/11
```
Substitute y = 30/11 into the first equation:
```
2x + 3(30/11) = 12
2x + 90/11 = 12
2x = 12 - 90/11
2x = (132 - 90)/11
2x = 42/11
x = 21/11
```
So, the intersection point is (21/11, 30/11).
Intersection of 2x + 3y = 12 and y = 0

Substitute y = 0 into the equation 2x + 3y = 12:
```
2x + 3(0) = 12
2x = 12
x = 6
```
So, the intersection point is (6, 0).
Intersection of 5x + 2y = 15 and x = 0

Substitute x = 0 into the equation 5x + 2y = 15:
```
5(0) + 2y = 15
2y = 15
y = 15/2
```
So, the intersection point is (0, 15/2).

The Vertices of the Feasible Region

Therefore, the vertices of the feasible region are:

(21/11, 30/11)
(6, 0)
(0, 15/2)

These vertices represent the corners of the feasible region and are essential for solving linear programming problems. The process of calculating these vertices involves solving systems of linear equations, which can be done using various algebraic techniques. The intersection of the two main constraint lines, 2x + 3y = 12 and 5x + 2y = 15, required a slightly more involved calculation, but by using the method of elimination, we were able to find the exact coordinates of the intersection point. The intersections with the axes were simpler to calculate, as they involved substituting either x = 0 or y = 0 into the equations. By finding these vertices, we have identified the key points that define the feasible region and are crucial for optimization. The coordinates of these vertices can now be used to evaluate an objective function and find the optimal solution for a linear programming problem.

Checking the Feasible Region

It's important to verify that these vertices indeed form the boundaries of the feasible region. This can be done by graphing the inequalities and visually confirming that the calculated vertices lie on the corners of the feasible region. Additionally, we can substitute the coordinates of the vertices back into the original inequalities to ensure that they satisfy all the constraints. This step is crucial to ensure that the calculated vertices are indeed valid and that we have not made any errors in our calculations. For example, we can substitute the coordinates of the vertex (21/11, 30/11) into the inequalities: 2(21/11) + 3(30/11) = 42/11 + 90/11 = 132/11 = 12, which satisfies the first constraint. Similarly, 5(21/11) + 2(30/11) = 105/11 + 60/11 = 165/11 = 15, which satisfies the second constraint. The non-negativity constraints are also satisfied as both x and y are positive. By performing these checks for all the vertices, we can be confident that we have correctly identified the boundaries of the feasible region. This verification process is an essential part of solving linear programming problems and ensures the accuracy of our results.

Conclusion

In conclusion, the vertices of the feasible region for the given constraints are (21/11, 30/11), (6, 0), and (0, 15/2). These points are critical for solving linear programming problems and finding optimal solutions. Understanding how to determine the vertices of a feasible region is a fundamental skill in mathematical optimization. This article has provided a detailed explanation of the process, including the graphical representation, the identification of potential vertices, and the algebraic calculations required to find their exact coordinates. By following these steps, you can confidently tackle a wide range of optimization challenges and make informed decisions in various fields. The vertices represent the extreme points of the feasible region, and they are crucial for finding the optimal solution to a linear programming problem. The process of identifying and calculating these vertices involves both graphical and algebraic techniques, which have been thoroughly explained in this article. By mastering these techniques, you will be well-equipped to solve complex optimization problems and apply them to real-world scenarios. The ability to determine the vertices of a feasible region is a valuable skill in fields such as operations research, economics, and engineering, where optimization is a key component of decision-making.

The vertices of the feasible region are (21/11, 30/11), (6, 0), and (0, 15/2).