Problem 6 - 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. $$ \begin{array}{l}{x+2 y \leq 6} \\ {2 x-y \leq 7} \\ {x \geq-2, y \geq-3}\end{array} $$

Step-by-Step Solution

Answer

Vertices: (5, 0.5), (12, -3), (-2, -11), (-2, -3). Maximum and minimum depend on the specific function values.

Step 1: Identify the Inequalities

We have the following inequalities: 1. \( x + 2y \leq 6 \)2. \( 2x - y \leq 7 \)3. \( x \geq -2 \)4. \( y \geq -3 \). These inequalities form the boundaries of the feasible region we need to graph.

Step 2: Graph the Boundary Lines

Graph the lines, for each inequality's equation set as equality:- For \( x + 2y = 6 \), intercepts are \((6,0)\) and \((0,3)\).- For \( 2x - y = 7 \), intercepts are \((3.5,0)\) and \((0,-7)\).- The lines \( x = -2 \) and \( y = -3 \) are vertical and horizontal lines, respectively.

Step 3: Determine the Feasible Region

The feasible region is determined by the overlap of the half-planes defined by the inequalities. For each line, select the correct side of the boundary:- Test \((0,0)\) for \( x + 2y \leq 6 \): \(0 + 0 \leq 6\), true.- Test \((0,0)\) for \( 2x - y \leq 7 \): \(0 - 0 \leq 7\), true.- Combine these regions with the constraints \( x \geq -2 \) and \( y \geq -3 \).

Step 4: Find Intersection Points

Calculate the vertices of the feasible region using substitution or elimination method:- Solve \( x + 2y = 6 \) and \( 2x - y = 7 \) to get the intersection \((5, 0.5)\).- From \( x + 2y = 6 \) and \( y = -3 \), solution is \((12, -3)\).- From \( 2x - y = 7 \) and \( x = -2 \), solution is \((-2, -11)\).- Given inequalities \( x = -2 \) and \( y = -3 \) directly yield the point \((-2, -3)\).

Step 5: Determine Minimum and Maximum Values

Calculate the values for the function on the vertices. Suppose the function is \( f(x, y) = ax + by \), the exact function was not provided:- \( f(5, 0.5) \)- \( f(12, -3) \)- \( f(-2, -11) \)- \( f(-2, -3) \)Evaluate each to find minimum and maximum values of the function over the feasible region.

Key Concepts

Graphing Inequalities

Graphing inequalities is a crucial step in solving systems of inequalities. It involves plotting each inequality on a coordinate plane. To begin, replace the inequality sign with an equal sign to form a boundary line. For example, each inequality such as \( x + 2y \leq 6 \) and \( 2x - y \leq 7 \) transforms into boundary lines: \( x + 2y = 6 \) and \( 2x - y = 7 \).

• Boundary Lines: Calculate the x and y intercepts to draw the boundary lines.
• Slope and Intercepts: Knowing where these lines intersect the axes can efficiently help draw them on a graph.

It's important to remember these lines split the plane into half-planes. One half-plane will satisfy the inequality, and the other will not. Test a point, typically \((0, 0)\), to determine which side of the boundary is part of the solution set. Use dashed lines when the inequality involves a \(<\) or \(>\), and solid lines for \(\leq\) or \(\geq\).

Feasible Region

Once you've graphed the individual inequalities, the feasible region is the area where all the solutions intersect. It's a visual representation of all the possible solutions that simultaneously satisfy all inequalities in the system. In the given problem, after graphing the inequalities, look for the region where all half-planes overlap.

Ultra-important aspects of the feasible region include:

• Shared Area: It's the common area that lies under each boundary of the inequalities.
• Visibility: This region is typically visible inside or can be defined as a polygon bounded by the intersection points.

Don’t forget to shade this area lightly to emphasize it while ensuring you maintain clarity regarding the intersection boundaries.

Vertices of Feasible Region

The vertices, or corner points, of the feasible region are where the boundaries of different inequalities intersect. These points are crucial as they often determine the maximum and minimum values of an objective function within the feasible region.

Ways to find vertices include:

• Intersection Points: Solve the set of equations (from boundaries) to find these corners.
• Vertex Importance: In optimization problems, check these points to find the greatest or smallest value of the function.

For example, intersections of the boundary lines in our problem occur at points like \((5, 0.5)\), \((12, -3)\), \((-2, -11)\), and \((-2, -3)\). Check these intersections critically, ensuring accuracy in calculations because errors here will directly affect later analysis.

Inequality Intersections

Inequality intersections refer to the points where the boundaries of different inequalities meet. In systems of inequalities, these intersections can only be determined by solving equations simultaneously. This might be through the substitution or elimination method.

Key elements of solving intersection problems:

• Algebraic Techniques: Practice both substitution and elimination to be ready for various types of equations.
• Precise Calculations: A minor error can misplace an entire feasible region on your graph.

For instance, solving \( x + 2y = 6 \) and \( 2x - y = 7 \) gives us the important point \((5, 0.5)\). Identifying all intersections will help create a precise feasible region and achieve correct evaluations of the functions related to these points. Accurate graphing leads to more reliable problem-solving.