Problem 34 - College Algebra Essentials

Question

Consider the objective function \(z=A x+B y \quad(A>0\) and \(B>0\) ) subject to the following constraints: \(2 x+3 y \leq 9, x-y \leq 2, x \geq 0,\) and \(y \geq 0 .\) Prove that the bbjective function will have the same maximum value at the vertices \((3,1)\) and \((0,3)\) if \(A=\frac{2}{3} B\)

Step-by-Step Solution

Answer

The provided claim is true. If the objective function \(z=Ax+By\) attains the same maximum value at both points \((3,1)\) and \((0,3)\), that indeed means that the coefficient \(A\) is \(\frac{2}{3}\) of \(B\), i.e., \(A= \frac{2}{3}B \).

Step 1: Substitute the vertices into the objective function

Begin by plugging the vertices into the objective function \(z\). For vertex \((3,1)\), the function becomes \(z = 3A + B\). For vertex \((0,3)\), \(z= 0A + 3B\) or simply \(z=3B\).

Step 2: Equate the two expressions for z

Since the optimal value of the objective function is the same at these two vertices, we can equate the expressions obtained in step 1: so, \(3A + B = 3B\).

Step 3: Solve for A

From the last equation, rearrange terms and solve for \(A\): \(3A = 3B - B\). Simplifying that gives \(3A = 2B\), which simplifies further to \(A= \frac{2}{3}B \), confirming the required result.

Key Concepts

Objective Function

In linear programming, the objective function is a formula that needs to be optimized, either maximized or minimized, depending on the problem's requirements. Think of it as the main goal you want to achieve using the resources available.
In our exercise, the objective function is given by the equation:

• \( z = Ax + By \)

Here, \( A \) and \( B \) are coefficients that determine the contribution of the variables \( x \) and \( y \) to the overall value of \( z \). The purpose is to find the maximum value of \( z \), subject to a set of constraints.
To find the maximum value effectively, we plug in different feasible solutions (which are the vertices of the feasible region) into this equation and compare the outcomes. The optimal solution will occur at one of these vertices.

Constraints

Constraints in linear programming set the boundaries within which the objective function must operate. They are like the rules that need to be followed to find a realistic solution.
For our problem, the constraints are:

• \( 2x + 3y \leq 9 \)
• \( x - y \leq 2 \)
• \( x \geq 0 \)
• \( y \geq 0 \)

These constraints form a feasible region when graphed, within which the solution for the objective function is valid.
The purpose of each constraint is to limit the values that \( x \) and \( y \) can take. This means that any potential solution (for example, a vertex) has to satisfy all these conditions to be considered viable. Thus, evaluating these constraints is crucial in narrowing down the feasible points where the maximum or minimum of the objective function might occur.

Vertices

Vertices are crucial in linear programming since they represent potential solutions to the optimization problem. They are the corner points of the feasible region formed by the constraints.
For this exercise, the critical vertices are \((3,1)\) and \((0,3)\). These points are determined by either intersecting the constraint lines or lying on the axes where constraints dictate. By evaluating the objective function at these vertices, we identify the maximum or minimum values it can achieve.
Why vertices matter:

• They simplify the process of finding the optimal solution.
• Theorems in linear programming indicate that if a maximum or minimum value exists for the objective function, it will occur at one of these vertices.

Therefore, evaluating the objective function at these key points provides a straightforward path to finding optimal solutions without calculating values inside the entire region.

Linear Equations

Linear equations form the foundation of both the constraints and the objective function in linear programming problems. They are expressions that plot straight lines when graphed.
In our exercise, the constraints

• \( 2x + 3y \leq 9 \) and \( x - y \leq 2 \)

are linear equations representing boundaries in the feasible region. The variables in these equations are usually plots or points on a coordinate plane.
Characteristics of linear equations:

• Each variable is raised to the power of 1, reflecting their linearity.
• Solutions of linear equations, when considered along with inequalities (constraints), help delineate a clear and bounded feasible region.

These regions and boundaries help dictate the scope within which the objective function must be optimized, guiding us to effective points of evaluation, typically at the vertices.