Question

Maximize the objective function subject to the constraints \(3 x+y \leq 15,4 x+3 y \leq 30\) \(x \geq 0\), and \(y \geq 0\) $$z=5 x+y$$

Short answer

The solution to this problem would be found by graphing the feasible region defined by the constraints, and then evaluating the objective function at each vertex of the feasible region. The maximum value of the function would give the required values of x and y.

Step-by-step solution

Plotting the Constraints

The first step is to plot the constraints provided on a graph. These constraints include \(3x + y \leq 15\), \(4x + 3y \leq 30\), \(x \geq 0\), and \(y \geq 0\). These inequalities will create a region of possible solutions on the graph.

Identifying the Feasible Region

The feasible region is the set of all points that satisfy all the constraints at the same time. The feasible region for this problem is a polygon on the xy-plane. The corners of the feasible region are known as vertices, and they correspond to possible solutions to the problem. These vertices can be found out by intersecting the lines representing the constraints

Maximizing the Objective Function

Once the feasible region is identified, the next step is to determine which point in this region maximizes the objective function \(z = 5x +y\). This can be done by substituting the coordinates of the vertices into the objective function and finding the maximum value.

Key concepts

Objective Function

In linear programming, the objective function represents what we want to accomplish, usually in the form of maximization or minimization. It is a mathematical expression that summarizes the goal of the problem. In our existing exercise, the objective function is given as: \[ z = 5x + y \] This equation states that our aim is to maximize the value of \(z\). Here, \(z\) depends on values of \(x\) and \(y\). These variables often represent quantities such as profits, costs, or resources. The coefficients 5 and 1 (from \(5x\) and \(y\)) indicate the weight or importance of \(x\) and \(y\) in achieving the goal. To solve the problem, we need to find the highest possible value for \(z\) by varying \(x\) and \(y\) within the allowed limits.

Feasible Region

The feasible region in a linear programming problem defines all possible points that satisfy the given constraints. It is the space where all the conditions of the problem meet. This region is often visualized on a graph as a polygon known as the "solution space." When graphing the inequalities: - \(3x + y \leq 15\) - \(4x + 3y \leq 30\) - \(x \geq 0\) - \(y \geq 0\) You will identify the area where all these conditions overlap. This overlap forms the feasible region. The vertices (or corners) of this polygon are crucial because they represent possible optimal solutions. For maximizing the objective function, check each vertex to find which one gives the highest value of \(z\). Exploring this region helps in identifying the optimal solution while considering all constraints.

Constraints

Constraints are the limitations or conditions that define how to solve a linear programming problem. They ensure that the solution is practical and within the set boundaries. In the given example, the constraints are expressed as inequalities:

• \(3x + y \leq 15\)
• \(4x + 3y \leq 30\)
• \(x \geq 0\)
• \(y \geq 0\)

These constraints form the basis for constructing the feasible region. The first two constraints are linear inequalities in two variables, which limit the combinations of \(x\) and \(y\). The last two constraints ensure that \(x\) and \(y\) are non-negative, often a necessary condition in practical scenarios where negative quantities wouldn't make sense (like negative profits, resources, etc.). Understanding and plotting these mathematical conditions help in navigating the feasible region to find optimal solutions.

Maximization

Maximization is the process of finding the highest possible value of the objective function. In linear programming, this is a common goal, especially in applications like profit or efficiency maximization. After establishing the objective function and constraints, follow these steps to achieve maximization:

• Identify the feasible region by plotting the constraints.
• Locate the vertices of this region.
• Substitute these vertices into the objective function.
• Compare the results and choose the vertex that yields the highest value of the objective function.

In our exercise, to find the optimal \(z\), you check the feasible region's vertices for their \(x\) and \(y\) values. By substituting these into \(z = 5x + y\), the vertex offering the highest outcome determines the solution. Thus, maximization provides a pathway to achieving the most favorable outcome within the given limits.