Problem 1 - Algebra 2

Question

Graph each system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region. $$ \begin{array}{l}{y \geq 2} \\ {x \geq 1} \\ {x+2 y \leq 9} \\ {f(x, y)=2 x-3 y}\end{array} $$

Step-by-Step Solution

Answer

Maximum value is 4 at (5, 2); minimum value is -10 at (1, 4).

Step 1: Graph the Inequality

Start by graphing each of the inequalities. For \( y \geq 2 \), draw a horizontal line at \( y = 2 \); shade the region above. For \( x \geq 1 \), draw a vertical line at \( x = 1 \); shade the region to the right. Finally, for \( x + 2y \leq 9 \), rearrange the equation to find the line \( y = -\frac{1}{2}x + \frac{9}{2} \). This line should be shaded below the line. The feasible region is where all these shaded areas intersect.

Step 2: Find Intersection Points

Determine where the lines intersect, as these points help to define the vertices of the feasible region. \( y = 2 \) intersects with \( x + 2y = 9 \) when \( 2 = -\frac{1}{2}x + \frac{9}{2} \). Solve for \( x \), giving \( x = 5 \). Thus, one vertex is (5, 2). Similarly, find the intersection of \( x = 1 \) with \( x + 2y = 9 \). Substitute \( x = 1 \) into \( y = -\frac{1}{2}x + \frac{9}{2} \), giving \( y = 4 \). Hence, the vertex is (1, 4). The intersection of \( y = 2 \) with \( x = 1 \) gives the vertex (1, 2). The vertices of the feasible region are (1, 2), (1, 4), and (5, 2).

Step 3: Evaluate the Objective Function at Each Vertex

Compute the value of \( f(x, y) = 2x - 3y \) at each vertex of the feasible region: - At (1, 2): \( f(1, 2) = 2(1) - 3(2) = 2 - 6 = -4 \)- At (1, 4): \( f(1, 4) = 2(1) - 3(4) = 2 - 12 = -10 \)- At (5, 2): \( f(5, 2) = 2(5) - 3(2) = 10 - 6 = 4 \)

Step 4: Determine Maximum and Minimum Values

Compare the values calculated in the previous step. The maximum value of the function \( f(x, y) \) is 4 at the vertex (5, 2). The minimum value is -10 at the vertex (1, 4).

Key Concepts

System of Inequalities

Linear programming often involves a set of inequalities known as a system of inequalities. These inequalities establish boundaries within which a solution must fall. In our example, the system is composed of three inequalities:

• \( y \geq 2 \)
• \( x \geq 1 \)
• \( x + 2y \leq 9 \)

Each inequality represents a half-plane in a coordinate system. These half-planes are defined by drawing a solid line for the equations and shading the appropriate region as prescribed by the inequality sign to indicate where solutions are allowed.
The solution to a system of inequalities is graphically represented in a coordinate system, where the shaded areas of all inequalities overlap and form the feasible region.

Feasible Region

When you graph a system of inequalities, the feasible region is the area where the shaded portions intersect. This region represents all possible solutions to the system of inequalities.
In our scenario, this is where the following boundaries meet:

• The horizontal line at \( y = 2 \)
• The vertical line at \( x = 1 \)
• The line expressed as \( y = -\frac{1}{2}x + \frac{9}{2} \)

The feasible region contains all possible (x, y) pairs that satisfy each inequality simultaneously. It's like a sweet spot where all conditions are met. Finding where this region lies is essential for further steps in linear programming, such as optimizing an objective function.

Objective Function

In linear programming, the objective function is the mathematical expression that you aim to maximize or minimize within the feasible region. It represents a criterion or goal that must be optimized. For this task, the objective function is defined as:

• \( f(x, y) = 2x - 3y \)

The objective is to find the highest and lowest values of this function within the feasible region. Solving this problem involves evaluating the function at different points (vertices) of the feasible region. The values obtained guide you in determining the optimum solution based on the context of the problem.

Vertex

After graphing the system of inequalities, the feasible region often takes the shape of a polygon. The intersections of the lines form what we call vertices. These points are crucial because they represent the candidates for optimal values of the objective function.
In our example, the vertices of the feasible region are:

• (1, 2)
• (1, 4)
• (5, 2)

These vertices are found by solving pairs of linear equations from the system of inequalities. Evaluating the objective function at these vertices allows us to determine possible maximum or minimum values.

Maximum and Minimum Values

Once the vertices of the feasible region have been identified, the next step is to evaluate the objective function at each vertex. This calculation yields potential maximum and minimum values of the function.

• At (1, 2): \( f(1, 2) = -4 \)
• At (1, 4): \( f(1, 4) = -10 \)
• At (5, 2): \( f(5, 2) = 4 \)

The maximum value of the objective function is 4, achieved at the vertex (5, 2). The minimum value is -10, obtained at the vertex (1, 4). Limits or endpoints of the feasible region are where these extreme values generally occur, demonstrating the importance of vertices in such analyses.