Problem 11 - College Algebra Essentials

Question

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints \(z=4 x+2 y\) \(\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\ 2 x+3 y \leq 12 \\ 3 x+2 y \leq 12 \\ x+y \geq 2\end{array}\right.\)

Step-by-Step Solution

Answer

The maximum value of the objective function is 20, occurring at \(x=4\) and \(y=2\).

Step 1: Graph the inequalities

The first step is to graph each of the inequalities: \(x \geq 0\), \(y \geq 0\), \(2 x+3 y \leq 12\), \(3 x+2 y \leq 12\), \(x+y \geq 2\). This will create a bounded region with vertices that represent possible solutions to the inequalities. You can use a graphing calculator or graph paper to do this.

Step 2: Identify the vertices of the region

By analyzing the graph, the vertices (corners) can be identified. These are the points where the lines represented by the inequalities intersect. In this scenario, they are (0,2), (2,0), (2,4), and (4,2).

Step 3: Evaluate the objective function at each vertex

The next step is to substitute each pair of (x, y) values representing the vertices into the objective function \(z=4 x+2 y\). This will give a value of z for each vertex of the region. Thus, the calculations are as follows:\n\nAt (0,2): \(z = 4*0 + 2*2 = 4\)\nAt (2,0): \(z = 4*2 + 2*0 = 8\)\nAt (2,4): \(z = 4*2 + 2*4 = 16\)\nAt (4,2): \(z = 4*4 + 2*2 = 20\)

Step 4: Determine the maximum value of the objective function and corresponding x, y values

The largest value of z calculated for the vertices of the graphed region represents the maximum value of the objective function. In this scenario, the maximum value of z is 20, which occurs for \(x=4\) and \(y=2\).

Key Concepts

Objective Function

In the realm of linear programming, the objective function is a mathematical expression that represents the goal of an optimization problem. It's essentially what you are trying to maximize or minimize through a set of variables subjected to certain constraints. In the given exercise, the objective function is defined as
\( z = 4x + 2y \).

This formula tells us that for any values of the variables \(x\) and \(y\), we can calculate the value of \(z\), which in this context, could be related to profit, cost, or any other quantity of interest that needs to be optimized. The coefficients of variables, such as 4 and 2 in the example, indicate the rate at which \(z\) changes concerning each variable.

System of Linear Inequalities

A system of linear inequalities comprises multiple linear inequalities that collectively define the feasible set of solutions for a linear programming problem. Unlike equations, inequalities don't express parity but instead specify a range of values, using signs like '<', '>', '\(\leq\)', or '\(\geq\)'. In the provided exercise, the system consists of the following inequalities:
\[\begin{align*}x &\geq 0,\ y &\geq 0,\ 2x + 3y &\leq 12,\ 3x + 2y &\leq 12,\ x + y &\geq 2.\end{align*}\]
These inequalities represent constraints that limit the values of \(x\) and \(y\), shaping the area in which the optimal solution can be found. The various inequalities in the system can also reflect different kinds of restrictions such as available resources, production capacities, budgetary limits, etc.

Graphing Inequalities

The concept of graphing inequalities is an integral part of visualizing the feasible region in linear programming problems. It involves plotting the system of inequalities on a coordinate plane and identifying the area where all the inequalities overlap, known as the feasible region.

To graph an inequality, you first treat it as an equation and plot the corresponding line. The inequality symbol indicates whether to shade above or below this line. For example, an inequality with '\(\leq\)' would mean shading below the line since the solutions are less than or equal to the boundary. The intersection of shadings from all inequalities reveals the solution set. It is vital to accurately graph these lines and their shadings, as any error here can lead to a miscalculation in finding the optimal solution. The vertices of this feasible region are particularly important as they are potential candidates for the optimal solution per the Fundamental Theorem of Linear Programming.

Optimization in Algebra

When we talk about optimization in algebra, we refer to finding the maximum or minimum values that a particular function can achieve within a given set of constraints. It's the crux of what linear programming aims to accomplish. In practical terms, optimization might mean maximizing profits, minimizing costs, or achieving the best outcome within the available resources.

In the context of linear programming, optimization is carried out by evaluating the objective function at the vertices of the feasible region obtained from graphing inequalities. Since linear programming deals with linear functions and systems, these optimal values are always found at the vertices, which is supported by linear programming's corner-point theorem. This principle greatly simplifies the search for an optimal solution, essentially telling us that we only need to check the corner points of the feasible region rather than every possible solution. The highest or lowest value of the objective function at these points will be the optimal solution, depending on whether we are maximizing or minimizing.