Finding The Minimum Value Of C = 7x + 8y With Constraints

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In the realm of linear programming, optimization problems often require us to find the minimum or maximum value of an objective function, subject to a set of constraints. These constraints, typically expressed as linear inequalities, define a feasible region within which the optimal solution lies. This article delves into a specific problem of this nature, aiming to determine the minimum value of the objective function C = 7x + 8y, given a system of inequalities that constrain the values of x and y. Understanding the methodology to solve such problems is crucial in various fields, including economics, engineering, and operations research, where resource allocation and optimization are paramount.

The problem at hand is a classic example of how mathematical techniques can be applied to real-world scenarios. The objective function C = 7x + 8y represents a quantity we wish to minimize, potentially cost, resource usage, or any other relevant metric. The constraints, expressed as inequalities, represent limitations or requirements that must be satisfied. For instance, 2x + y ≥ 8 and x + y ≥ 6 could represent minimum production levels or resource availability, while x ≥ 0 and y ≥ 0 ensure that the variables are non-negative, a common condition in many practical applications. Solving this problem involves a combination of algebraic manipulation, graphical analysis, and logical deduction. We will explore how to graph the feasible region defined by the constraints, identify the corner points of this region, and evaluate the objective function at these points to determine the minimum value. This process not only provides the solution but also enhances our understanding of the interplay between constraints and optimization in linear programming problems. Let's embark on a journey to unlock the solution to this fascinating problem, step by step, ensuring a comprehensive understanding for both novices and seasoned problem-solvers.

Our primary objective is to find the minimum value of the function C, defined as C = 7x + 8y. This function is subject to the following constraints, which are inequalities that limit the possible values of x and y:

  1. 2x + y ≥ 8
  2. x + y ≥ 6
  3. x ≥ 0
  4. y ≥ 0

These constraints define a region in the xy-plane known as the feasible region. The feasible region encompasses all points (x, y) that simultaneously satisfy all the given inequalities. The minimum value of C will occur at one of the corner points (vertices) of this feasible region. Therefore, our task is to first identify the feasible region by graphing the inequalities and then determine its corner points. Once we have the coordinates of these corner points, we can evaluate the objective function C = 7x + 8y at each point and select the one that yields the smallest value. This process embodies the fundamental principle of linear programming: the optimal solution (minimum or maximum) always lies at a vertex of the feasible region. Understanding this principle is crucial for solving a wide range of optimization problems, from resource allocation to production planning. The constraints x ≥ 0 and y ≥ 0 restrict our solutions to the first quadrant of the coordinate plane, which simplifies the graphing process. The other two inequalities, 2x + y ≥ 8 and x + y ≥ 6, define lines that bound the feasible region. The region will be the area that satisfies all these inequalities simultaneously. Our next step is to graph these inequalities, identify the feasible region, and pinpoint its corner points. This graphical approach provides a visual representation of the problem, making it easier to understand and solve. Let's move on to the graphing process and uncover the feasible region that holds the key to our solution.

To visually represent the constraints, we will graph each inequality on the xy-plane. This process involves converting each inequality into its corresponding equation, plotting the line, and then determining which side of the line represents the solution set for the inequality.

  1. 2x + y ≥ 8: To graph this, we first consider the equation 2x + y = 8. We can find two points on this line by setting x = 0 and solving for y (which gives us the point (0, 8)), and setting y = 0 and solving for x (which gives us the point (4, 0)). Plotting these points and drawing a line through them gives us the line 2x + y = 8. Since the inequality is 2x + y ≥ 8, we need to determine which side of the line represents the solutions. We can test a point, such as (0, 0), in the inequality. Since 2(0) + 0 ≥ 8 is false, the region that does not contain (0, 0) is the solution region. Thus, we shade the region above the line.

  2. x + y ≥ 6: Similarly, we consider the equation x + y = 6. Setting x = 0 gives y = 6 (the point (0, 6)), and setting y = 0 gives x = 6 (the point (6, 0)). Plotting these points and drawing a line through them gives us the line x + y = 6. Testing the point (0, 0) in the inequality x + y ≥ 6, we find that 0 + 0 ≥ 6 is false. Therefore, we shade the region above the line.

  3. x ≥ 0: This inequality represents all points to the right of the y-axis, so we shade the region to the right of the y-axis.

  4. y ≥ 0: This inequality represents all points above the x-axis, so we shade the region above the x-axis.

The feasible region is the area where all shaded regions overlap. This region is a polygon, and its vertices (corner points) are the points where the boundary lines intersect. Identifying these vertices is crucial because the minimum value of the objective function will occur at one of these points. The graphical representation provides a clear visual aid in identifying the feasible region and its vertices, making it easier to determine the optimal solution. The next step is to find the coordinates of these vertices, which will involve solving systems of linear equations. Understanding the graphical representation is essential for grasping the concept of feasible regions and their role in optimization problems. Let's proceed to the next step and determine the coordinates of the vertices of our feasible region.

The corner points, also known as vertices, of the feasible region are the points where the boundary lines intersect. These points are crucial because the minimum value of the objective function C = 7x + 8y will occur at one of these corners. From the graph, we can identify three corner points:

  1. Intersection of 2x + y = 8 and the y-axis (x = 0): Substituting x = 0 into 2x + y = 8, we get 2(0) + y = 8, which simplifies to y = 8. Thus, the corner point is (0, 8).

  2. Intersection of x + y = 6 and the x-axis (y = 0): Substituting y = 0 into x + y = 6, we get x + 0 = 6, which simplifies to x = 6. Thus, the corner point is (6, 0).

  3. Intersection of 2x + y = 8 and x + y = 6: To find this intersection, we need to solve the system of equations:

    • 2x + y = 8
    • x + y = 6

    We can subtract the second equation from the first to eliminate y: (2x + y) - (x + y) = 8 - 6, which simplifies to x = 2. Substituting x = 2 into x + y = 6, we get 2 + y = 6, which simplifies to y = 4. Thus, the corner point is (2, 4).

Now that we have identified all the corner points of the feasible region, we have a set of candidate points for the minimum value of the objective function. These points represent the extreme values of the feasible region, and one of them will yield the optimal solution. The process of finding these corner points involves algebraic manipulation and solving systems of linear equations, skills that are fundamental in linear programming. Understanding how these points are derived is essential for applying the corner-point method effectively. The next step is to evaluate the objective function C = 7x + 8y at each of these corner points to determine which one yields the minimum value. Let's proceed to this evaluation and unveil the solution to our problem.

Now that we have identified the corner points of the feasible region, we can evaluate the objective function C = 7x + 8y at each of these points to determine the minimum value. The corner points we found are (0, 8), (6, 0), and (2, 4). We will substitute the x and y coordinates of each point into the objective function and calculate the corresponding value of C.

  1. At (0, 8):

    • C = 7(0) + 8(8)
    • C = 0 + 64
    • C = 64

  2. At (6, 0):

    • C = 7(6) + 8(0)
    • C = 42 + 0
    • C = 42

  3. At (2, 4):

    • C = 7(2) + 8(4)
    • C = 14 + 32
    • C = 46

By evaluating the objective function at each corner point, we have obtained the following values for C: 64 at (0, 8), 42 at (6, 0), and 46 at (2, 4). The minimum value of C is the smallest of these values. Comparing the values, we can see that the minimum value of C is 42, which occurs at the point (6, 0). This result demonstrates the core principle of the corner-point method: the optimal solution to a linear programming problem always occurs at a corner point of the feasible region. The evaluation process is straightforward, involving simple substitution and arithmetic. However, it is a crucial step in the solution process, as it allows us to pinpoint the exact point that yields the minimum (or maximum) value of the objective function. Understanding this process is key to mastering linear programming problems. Now that we have found the minimum value of C, we can conclude our analysis and present the final answer. Let's summarize our findings and provide the solution to the problem.

In summary, we sought to find the minimum value of the objective function C = 7x + 8y, subject to the constraints 2x + y ≥ 8, x + y ≥ 6, x ≥ 0, and y ≥ 0. To solve this linear programming problem, we first graphed the constraints to identify the feasible region, which is the area that satisfies all the given inequalities. The feasible region was a polygon, and its corner points were the points of intersection of the boundary lines.

We then identified the corner points of the feasible region. These points were found to be (0, 8), (6, 0), and (2, 4). To determine which of these points yields the minimum value of C, we evaluated the objective function at each corner point. This involved substituting the x and y coordinates of each point into the equation C = 7x + 8y and calculating the resulting value of C.

Upon evaluating the objective function, we found that:

  • At (0, 8), C = 64
  • At (6, 0), C = 42
  • At (2, 4), C = 46

Comparing these values, we concluded that the minimum value of C is 42, which occurs at the point (6, 0). This result is consistent with the principle of linear programming, which states that the optimal solution (minimum or maximum) always lies at a vertex of the feasible region. This process highlights the importance of graphical analysis in understanding the constraints and feasible region, as well as the algebraic techniques used to find the corner points and evaluate the objective function. The solution to this problem provides a clear illustration of how linear programming can be used to optimize a function subject to certain constraints, a technique widely applied in various fields such as economics, engineering, and operations research. We have successfully decoded the minimum value of C, providing a comprehensive solution to the given problem.