Problem 7 - Algebra and Trigonometry

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 $$ \begin{aligned} &z=4 x+y\\\ &\left\\{\begin{array}{l} {x \geq 0, y \geq 0} \\ {2 x+3 y \leq 12} \\ {x+y \geq 3} \end{array}\right. \end{aligned} $$

Step-by-Step Solution

Answer

The maximum value of the objective function \(z=4x+y\) is 12, which occurs at the point \((3,0)\).

Step 1: Graph the System of Inequalities

Start by sketching the two constraints, \(2x+3y \leq 12\) and \(x+y \geq 3\), in the first quadrant since \(x, y \geq 0\). The constraints' intersection forms a feasible region, which is a triangle in this case.

Step 2: Find the Values of Objective Function

Next, find the value of the objective function, \(z=4x+y\), at each corner of the feasible region. The corners are: \((0,0)\), \((0,4)\), and \((3,0)\). By substituting each corner's coordinates into the objective function, we get: \(z(0,0) = 0\), \(z(0,4) = 4\), and \(z(3,0) = 12\).

Step 3: Determine the Maximum Value of the Objective Function

Lastly, determine the maximum value of the objective function from the values found in step 2. The maximum value of z is 12, which occurs at the point \((3,0)\).

Key Concepts

Objective Function

In linear programming, the **objective function** is a key component as it defines what needs to be optimized—either maximized or minimized. It is usually expressed as a linear equation. The coefficients in this equation represent how much each decision variable contributes to the value that is being optimized. In our given exercise, the objective function is defined as \(z=4x+y\).
This indicates that for each unit increase in \(x\), the value of \(z\) increases by 4, and for each unit increase in \(y\), \(z\) increases by 1. To solve a problem effectively, understanding the interplay between the coefficients and the decision variables is crucial.
By finding the values of \(x\) and \(y\) that maximize or minimize \(z\), we achieve optimal results that meet the problem's objective.

Linear Inequalities

The constraints in linear programming problems are often represented by **linear inequalities**. These inequalities describe the conditions that the solution must satisfy. In this exercise, the constraints are:

• \(x \geq 0\)
• \(y \geq 0\)
• \(2x+3y \leq 12\)
• \(x+y \geq 3\)

These inequalities restrict the possible values for \(x\) and \(y\).
The first two inequalities ensure that both variables are non-negative, which is typical in real-world problems as we cannot have negative quantities.
Visualizing these conditions on a graph allows us to see where the solution lies. Graphing these inequalities helps identify the **feasible region**.

Feasible Region

In linear programming, the **feasible region** represents all possible points that satisfy the given constraints simultaneously. It is often a bounded area on the graph, formed at the intersection of all the linear inequalities. Using the constraints from our exercise, the feasible region lies within the first quadrant and is defined by the lines \(2x+3y \leq 12\) and \(x+y \geq 3\).
This region is usually polygonal, like a triangle or a quadrilateral. In this case, it is a triangle.
The corners, or vertices, of this polygon are significant because they are points of intersection of the constraints. Evaluating the objective function at these vertices helps in identifying which point gives the optimal solution.

Maximum Value

Once the feasible region is identified, the **maximum value** of the objective function can be determined. This involves calculating the value of the objective function at each vertex of the feasible region. In our example, the vertices are \((0,0)\), \((0,4)\), and \((3,0)\).
By plugging these values into the objective function \(z=4x+y\), we get:

• \(z(0,0) = 0\)
• \(z(0,4) = 4\)
• \(z(3,0) = 12\)

To find the maximum value, compare these results. The highest value is 12, which occurs at the vertex \((3,0)\).
This means that when \(x=3\) and \(y=0\), the objective function reaches its maximum value, satisfying all constraints effectively. This systematic approach ensures that the optimal solution is both valid and feasible.