Problem 64 - Algebra 2

Question

Graph each system of constraints. Find all vertices. Then find the values of \(x\) and \(y\) that maximize or minimize the objective function. $$ \left\\{\begin{aligned} x+2 y & \leq 8 \\ x & \geq 2 \\ y & \geq 0 \end{aligned}\right. $$ $$ \begin{array}{l}{\text { Minimize for }} \\ {C=x+3 y}\end{array} $$

Step-by-Step Solution

Answer

The values of \( x \) and \( y \) that minimize the objective function are \( x = 2 \), \( y = 3 \); with a minimum value for \( C \) of 11.

Step 1: Graph the inequalities

Plot the constraints on a coordinate plane. Start by turning each inequality into an equation: 1) Transform the first inequality, \( x + 2y = 8 \), into y-intercept form to graph it. 2) Draw a vertical line at \( x = 2 \) for the second inequality. 3) The third inequality, \( y g g g e 0 \), is represented by the x-axis. The area that satisfies all three inequalities is the feasible region.

Step 2: Identify the vertices of the feasible region

Find the points of intersection between the lines, which are the vertices of the feasible region. The vertices occur where two of the boundary lines intersect. Solve the equations algebraically to find these points or use the graph to estimate their locations.

Step 3: Calculate the objective function at the vertices

Calculate the value of the objective function \( C = x + 3y \) at each vertex of the feasible region. The minimum or maximum value of \( C \) will occur at one of these vertices.

Step 4: Determine the maximum or minimum value of the objective function

Compare the values of the objective function calculated in Step 3 to identify the vertex where \( C \) is minimized. The corresponding \( x \) and \( y \) values at this vertex give the solution to the problem.

Key Concepts

Linear Inequalities in Two Variables

Understanding linear inequalities in two variables is the foundation of analyzing systems of inequalities. A linear inequality looks much like a linear equation, with the inequality sign taking the place of equality. In the context of a system of inequalities, each linear inequality defines a region on the coordinate plane where all solutions to the inequality exist.

For example, take the inequality \(x + 2y \leq 8\). To graph this inequality, we first graph the line \(x + 2y = 8\) as if it were an equation. This line divides the plane into two halves. The inequality \(x + 2y \leq 8\) tells us that our region of interest is the set of points that lie on or below this line, since \(\leq\) includes the boundary line itself. It's essential to use a test point to determine which side of the line is part of the solution set. Any point that does not lie on the line can be used as a test point; a common choice is \(0,0\) if it is not on the line.

Feasible Region

The feasible region is the area where all constraints of a system overlap. In graphing a system of inequalities, each inequality shades a portion of the graph. The feasible region is the common area where all these shaded regions intersect, and it represents all the possible solutions to the system.

In our exercise, the system of inequalities consists of \(x + 2y \leq 8\), \(x \geq 2\), and \(y \geq 0\). Graphically, we find the feasible region by plotting each inequality as described above and then identifying where they overlap. The feasible region in our problem is a polygonal shape on the coordinate grid, bounded by the appropriate sides of the lines from the inequalities. It is critical for students to shade lightly or use different patterns for each inequality to ensure the feasible region is clear.

Objective Function Optimization

In a graphing system of inequalities problem, we are often asked to find the optimal value—either the maximum or the minimum—of a function, known as the objective function. This function typically relates to a real-world quantity we want to optimize, such as profit, cost, or distance.

In the case of our exercise, the objective function to be minimized is \(C = x + 3y\). To find the optimal point, we evaluate this function at each vertex of the feasible region that was previously determined. These vertices are the only candidates for the optimal solution in linear programming problems due to the nature of straight-line constraints and linear objectives. It's a process of calculation and comparison to identify which vertex gives the smallest value for \(C\) since we are minimizing in this example.

Vertices of a System of Constraints

Vertices play a crucial role when dealing with systems of linear inequalities. A vertex is a point where two or more boundary lines intersect. These points are paramount because, according to the corner-point principle, the optimal solution to a linear programming problem will always be found at a vertex of the feasible region if the optimal solution exists and is finite.

For our example, after graphing the inequalities and finding the feasible region, we identify where the boundary lines intersect to locate the vertices. Algebraic methods can be used to find the exact coordinates if the graph is not precise enough. For instance, we can set the equations of two boundary lines equal to one another to solve for the intersection points, which are our vertices. In this problem, we find vertices by looking at intersections of \(x + 2y = 8\) with \(x = 2\) and the x-axis, where \(y = 0\).