Problem 6 - College Algebra

Question

In Exercises 5–14, 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 $$\begin{aligned}&z=2 x+3 y\\\&\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\\2 x+y \leq 8 \\\2 x+3 y \leq 12\end{array}\right.\end{aligned}$$

Step-by-Step Solution

Answer

The maximum value of function \(z=2x+3y\) occurs at (3,2) with a z-value of 12.

Step 1: Graph the constraints

We first need to graph the constraints. They are: \(x \geq 0\), \(y \geq 0\), \(2x + y \leq 8\) and \(2x + 3y \leq 12\). Now associate the inequalities with boundary lines that are \(x = 0\), \(y = 0\), \(2x + y = 8\) and \(2x + 3y = 12\). After sketching all these lines appropriately, we take into consideration the inequality signs to find the area of the graph that satisfies all constraints.

Step 2: Identify the corner points

We find the corner points on the graph plotting the solution to the system of inequality equations. These occur where the boundary lines cross. The corner points obtained are (0,0), (0,4), (3,2) and (4,0). These points form the boundary of the feasible region.

Step 3: Compute the objective function at each corner

Plug in each of the corner points (0,0), (0,4), (3,2) and (4,0) into the objective function \(z=2x+3y\). This will provide respective z values at each corner of the feasible region.

Step 4: Find the maximum

Compare the values of \(z\) obtained in step 3 to identify the maximum. The values of \(x\) and \(y\) at this maximum z-value are the solution to this problem.

Key Concepts

Objective Function

In the realm of linear programming, the objective function is like the mission statement of the problem. It tells us what we're aiming to maximize or minimize. In our exercise, the objective function is given by the formula \(z = 2x + 3y\). This means that for any values of \(x\) and \(y\) you choose, you can calculate \(z\), which is called the objective value.

The coefficients 2 and 3 are critical because they determine how much each of the variables contributes to the value of \(z\). If you wanted to increase \(z\), you'd try to adjust \(x\) and \(y\) within the system's constraints to make \(z\) as large as possible. The objective function is central to solving optimization problems because it defines what we want to achieve.

System of Inequalities

The system of inequalities acts as a set of rules or boundaries for the problem. In this exercise, the constraints are:

• \(x \geq 0\)
• \(y \geq 0\)
• \(2x + y \leq 8\)
• \(2x + 3y \leq 12\)

Each of these inequalities limits the possible choices for \(x\) and \(y\). Think of them as both limitations and guides that help us home in on feasible solutions.
Graphically, these inequalities form lines on a graph. For example, \(x \geq 0\) and \(y \geq 0\) ensure that we only consider solutions in the first quadrant. The other inequalities create additional boundaries that the solution must respect. Together, these lines carve out a special area on the graph known as the feasible region.

Feasible Region

The feasible region is a crucial part of solving a linear programming problem. It is the part of the graph where all the inequalities overlap, forming a polygonal area.

In our exercise, the feasible region is defined by the lines \(x = 0\), \(y = 0\), \(2x + y = 8\), and \(2x + 3y = 12\). This region includes all the possible solutions to the system of inequalities where both \(x\) and \(y\) satisfy all the constraints.

It's important because the optimal solution to our objective function lies within this region. Only the points inside or on the boundary of this feasible region are considered when searching for the maximum or minimum value of the objective function.

Corner Points

The corner points are where the lines defining the feasible region intersect. In linear programming, they are significant because the maximum or minimum value of an objective function usually occurs at these points.

For our given problem, the corner points are found at (0,0), (0,4), (3,2), and (4,0). These are the intersections of the boundary lines of the feasible region.

To find the optimal solution for the objective function \(z = 2x + 3y\), we evaluate \(z\) at each of these corner points. The corner point that yields the highest (or lowest if minimizing) value of \(z\) will give us the solution to the problem. This is because in a linear problem, solutions are most effectively explored at these vertices due to the shape and nature of the polygons formed by constraints.