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.

Verified step by step guidance

1. Identify the objective function and the system of linear inequalities given in the problem. The objective function is usually in the form \(Z = ax + by\), where \(a\) and \(b\) are constants, and \(x\) and \(y\) are variables.
2. Graph each inequality on the coordinate plane. To do this, first convert each inequality to an equation by replacing the inequality sign with an equals sign, then plot the corresponding line. Determine which side of the line satisfies the inequality by testing a point (often the origin) and shade the appropriate region.
3. Find the feasible region by identifying the overlapping shaded area that satisfies all inequalities simultaneously. This region represents all possible solutions that meet the constraints.
4. Locate the corner points (vertices) of the feasible region. These points are where the boundary lines intersect. You can find these points by solving the system of equations formed by pairs of boundary lines.
5. Evaluate the objective function \(Z = ax + by\) at each corner point by substituting the \(x\) and \(y\) values of the vertices into the objective function. Compare these values to determine which one gives the maximum value, and note the corresponding \(x\) and \(y\) values.

Key Concepts

System of Linear Inequalities

A system of linear inequalities consists of two or more inequalities involving the same variables. Each inequality divides the coordinate plane into regions, and the solution to the system is the intersection of these regions. Graphing the system helps visualize the feasible region that satisfies all constraints simultaneously.

Feasible Region and Corner Points

The feasible region is the set of all points that satisfy the system of inequalities, often forming a polygon on the graph. The corner points (vertices) of this region are critical because, in linear programming, the maximum or minimum values of the objective function occur at these points.

Objective Function and Optimization

An objective function is a linear expression in variables (e.g., x and y) that we want to maximize or minimize. By evaluating this function at each corner point of the feasible region, we identify which point yields the optimal (maximum or minimum) value, solving the optimization problem.