Problem 5 - Thinking Mathematically

Question

In Exercises 5-8, 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 $$ z=x+y $$ Constraints $$ \left\\{\begin{array}{l} x \leq 6 \\ y \geq 1 \\ 2 x-y \geq-1 \end{array}\right. $$

Step-by-Step Solution

Answer

The maximum value of the objective function is 13 and it occurs at the point (6, 7).

Step 1: Graph the Inequalities

Plot the lines \(x = 6\), \(y = 1\), and \(2x - y = -1\) in Cartesian Coordinate system. Then, shaded the regions that meet all the rules of constraints. Namely, the area to the left of the \(x = 6\), above the \(y = 1\), and above the \(2x - y = -1\). This shaded region represents all the possible solutions.

Step 2: Identify the corner points

Find the intersection of the lines that bound the shaded area. The intersections are: (6,1), (6,7) and (3,1. These are the corner points of the feasible region.

Step 3: Calculate the value of the objective function

From the objective function \(z = x + y\), calculate the value of \(z\) at each of the intersection points. At (6,1), \(z = 6 + 1 = 7\), at (6,7), \(z = 6 + 7 = 13\), and at (3,1), \(z = 3 + 1 = 4\).

Step 4: Determine the maximum value

Compare the values of the objective function at the corner points. The maximum value of z is 13 at point (6, 7).

Key Concepts

Objective Function

In linear programming, the objective function is the crux of the problem you are trying to solve. It defines what you're aiming to maximize or minimize, such as profit, cost, or any measurable outcome. In this exercise, the objective function is given by \(z = x + y\).

This equation means that the goal is to find the point where the sum of the variables \(x\) and \(y\) is the greatest.

It's crucial to calculate the objective function at specific points to determine the optimal solution.

Feasible Region

The feasible region is a fundamental concept in linear programming. It represents all possible solutions that satisfy the set of given constraints. These constraints are often a system of linear inequalities.

For this exercise, the feasible region is formed by the area of the graph where all inequalities are true simultaneously. You find this by plotting each inequality's line, then shading all areas where the constraints overlap.

• The inequality \(x \leq 6\) includes points to the left of the line \(x = 6\).
• The inequality \(y \geq 1\) covers areas above the line \(y = 1\).
• The inequality \(2x - y \geq -1\) represents areas above the line \(2x - y = -1\).

By intersecting these regions, you get the feasible region that contains all potential solutions.

System of Linear Inequalities

A system of linear inequalities are constraints that restrict the possible solutions in a linear programming problem. Solving these inequalities helps outline the feasible region, which encompasses the possible values of \((x, y)\) that satisfy all constraints.

In the system provided:

• \(x \leq 6\) means \(x\) cannot be greater than 6.
• \(y \geq 1\) ensures \(y\) is always at least 1.
• \(2x - y \geq -1\) creates a more complex boundary by aligning \(x\) and \(y\) in a specific way.

Each inequality adds a slice of the possible solution space, limiting the regions where an answer can be found, hence forming the constraints.

Corner Points

Corner points of the feasible region are crucial as they often contain the optimal solution for the objective function in linear programming problems. They are where the boundary lines of the constraints intersect.

In this context, the corner points of the feasible region are found at the intersections \((6, 1)\), \((6, 7)\), and \((3, 1)\).

These points are evaluated in the objective function to determine which produces the maximum or minimum value. By computing \(z = x + y\) at each point, we discover which one reaches the highest number, leading us to the optimal solution.