Problem 10 - 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 \(z=5 x-2 y\) \(\left\\{\begin{array}{l}0 \leq x \leq 5 \\ 0 \leq y \leq 3 \\ x+y \geq 2\end{array}\right.\)

Step-by-Step Solution

Answer

The maximum of the objective function \(z = 5x - 2y\) occur at the vertices of the graphed region formed by the constraints. The exact coordinates and maximum value depend on the results from step 2.

Step 1: Graph the constraints

Start by graphing the following inequalities on the Cartesian plane: 1) \(0 \leq x \leq 5\): It forms a vertical strip between the lines \(x = 0\) and \(x = 5\).2) \(0 \leq y \leq 3\): Together with the first constraint it forms a rectangle in the first quadrant.3) \(x+y \geq 2\): This is a upward sloping line in the first quadrant which cuts the rectangle formed by the first two inequalities. The area above this line is the feasible region, limited by the rectangle.

Step 2: Find Objective Function values at the corners

The vertices of the feasible region are the points: (0,2), (2,2), (2,3), (5,3), (5,0). Plug these coordinates into the objective function \(z=5x-2y\) to get the values at the corners.

Step 3: Determine Maximum Value

Compare the objective function values calculated in Step 2. The maximum value of the objective function and the coordinates \((x, y)\) where this maximum occurs, corresponds to the highest calculated value.

Key Concepts

Objective Function

In linear programming, the objective function is a mathematical expression that you aim to maximize or minimize. This function is composed of variables and coefficients that represent the relationships between different elements. In the given problem, the objective function is expressed as \(z = 5x - 2y\). Here, \(x\) and \(y\) are variables, and \(5\) and \(-2\) are their respective coefficients. The goal is to find the values of \(x\) and \(y\) that will give us the maximum value for \(z\).The coefficients determine how much each variable impacts the objective function's value. A positive coefficient means increasing the variable will increase \(z\), while a negative coefficient means the opposite. Understanding the role of each part of the function helps identify how changes in \(x\) and \(y\) affect the overall value.

System of Inequalities

A system of inequalities provides the constraints for the objective function. These constraints define limitations on the variables, which in turn affect possible solutions for the problem. In this exercise, the system of inequalities is as follows:

• \(0 \leq x \leq 5\)
• \(0 \leq y \leq 3\)
• \(x + y \geq 2\)

Each inequality represents a condition that \(x\) and \(y\) must satisfy. The first two inequalities model limitations on the values of \(x\) and \(y\) within specific boundaries, forming a rectangular region on the graph. The third inequality adds another constraint that ensures the sum of \(x\) and \(y\) cannot be less than 2. By solving these inequalities together, we establish a feasible region where the objective function can be evaluated.

Graphing Inequalities

Graphing inequalities is the process of visually representing the constraints on a coordinate plane. Each inequality corresponds to a region on the graph:

• For \(0 \leq x \leq 5\), shade the vertical strip between the lines \(x = 0\) and \(x = 5\).
• For \(0 \leq y \leq 3\), shade horizontally between \(y = 0\) and \(y = 3\).
• \(x + y \geq 2\) introduces a line where points above this line are shaded. This line intersects with other constraints to cut through the previously shaded area.

Together, these shaded areas intersect to form a polygon, called the feasible region. By plotting each constraint, you can visually see where all conditions hold simultaneously.

Maximum Value

Finding the maximum value of an objective function involves evaluating its formula at the vertices of the feasible region. These vertices are intersection points of the graphed inequalities:

• (0, 2)
• (2, 2)
• (2, 3)
• (5, 3)
• (5, 0)

Insert each vertex into the objective function \(z = 5x - 2y\) to compute \(z\) at those points. Compare these values to find which provides the largest value of \(z\). In this case, the coordinates that yield the maximum value of \(z\) signify the optimal solution for the problem.

Feasible Region

The feasible region is the area on the graph where all inequalities overlap. This region contains all possible solution points that satisfy every constraint in the system of inequalities. To identify the feasible region, locate where the shaded areas from each inequality's graph intersect:

• It lies within the rectangle formed by \(0 \leq x \leq 5\) and \(0 \leq y \leq 3\).
• It extends above the line \(x + y = 2\), which cuts through the rectangle.

Only points within this intersection can be considered in evaluating the objective function, as they represent combinations of \(x\) and \(y\) that meet all given conditions. The boundary vertices of this region are tested to identify where the objective function achieves its optimum.