Problem 14 - 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 $$\begin{aligned}&z=5 x+6 y\\\&\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\\2 x+y \geq 10 \\\x+2 y \geq 10 \\\x+y \leq 10\end{array}\right.\end{aligned}$$

Step-by-Step Solution

Answer

The final solution can only be determined after performing these calculations. After executing this process, one can define the maximum value of z and the coordinates x and y, that produces this maximum value.

Step 1: Graph the inequalities

To visualize the problem, the inequalities need to be graphed first, which means plotting four different inequality regions represented by \(x \geq 0\), \(y \geq 0\), \(2x+y \geq 10\), \(x+2y \geq 10\), and \(x+y \leq 10\). This creates a region in the XY plane which is bound by these inequalities.

Step 2: Find the vertices

The vertices of the region are formed by points where any two lines intersect. Therefore, we need to find the solution set (x, y) for the systems of equations formed by each pair of lines, which gives us the corner points of the region.

Step 3: Substitute into the objective function

The next step is to substitute the vertices into the objective function \(z=5x+6y\). This gives us the value of the objective function at each corner point in the feasible region.

Step 4: Determine the maximum value

Finally, compare the values obtained in the previous step. The largest value among them is the maximum value of the objective function. The x and y value that produced this maximum value are the solutions to this problem.

Key Concepts

Objective Function

The core goal of linear programming is often expressed through an objective function. This function provides a mathematical expression that we wish to either maximize or minimize. In our exercise, the objective function is defined as \( z = 5x + 6y \). Here, \( z \) represents the value we're interested in maximizing. The coefficients of \( 5 \) and \( 6 \) detail how much each unit of \( x \) and \( y \) contributes to the overall value of \( z \). By solving the exercise, we are trying to find the combination of \( x \) and \( y \) that yields the highest possible value for \( z \). In practical scenarios, this could relate to maximizing profit, optimizing resources, or finding the best outcome among various possibilities. By focusing on the objective function, we're setting a clear target for our analysis and computations.

System of Inequalities

The system of inequalities outlines the constraints or limitations within which our solution must exist. They are expressed as linear inequalities in the form:

• \( x \geq 0 \)
• \( y \geq 0 \)
• \( 2x + y \geq 10 \)
• \( x + 2y \geq 10 \)
• \( x + y \leq 10 \)

These inequalities specify the conditions that \( x \) and \( y \) must satisfy. The first two, \( x \geq 0 \) and \( y \geq 0 \), ensure that \( x \) and \( y \) remain non-negative, restricting the feasible solutions to the first quadrant of the coordinate plane. The remaining inequalities define more specific linear relationships and boundaries within this quadrant. Together, they form a system that constrains our possible solutions. Solving the system involves identifying all points \( (x, y) \) that satisfy each inequality simultaneously.

Graphing Inequalities

Graphing linear inequalities is a key step in locating the feasible region. It involves plotting each boundary line of the inequality on a graph. For each inequality, you begin by treating the inequality as an equation (e.g., \( 2x + y = 10 \)) and graphing the line. Then determine which side of the line satisfies the inequality by testing a point not on the line, often the origin \((0, 0)\) unless it lies on the line:

• For \( 2x + y \geq 10 \), the region above the line will satisfy the inequality.
• For \( x + 2y \geq 10 \), shade the area above its line.
• For \( x + y \leq 10 \), you'll shade below the line.

The solution to the system of inequalities is where these shaded regions intersect. A clear graph shows this intersection, helping us visually locate the feasible region.

Feasible Region

The feasible region is the overlapping area on the graph where all the constraints are met. It represents all possible solutions that satisfy the system of inequalities. After plotting each individual inequality, the intersection of these areas results in a polygon-shaped region. This is the essence of the linear programming problem, as it contains all the points \((x, y)\) that are potential solutions within the constraints.Within this feasible region are several crucial points known as vertices or corner points. These vertices are formed where the lines of the inequalities intersect. Evaluating the objective function at each of these vertices is critical, as the maximum or minimum value of the objective function will occur at one of these points. By comparing the values of the objective function at these vertices, we can identify the optimal solution for the problem at hand.