Problem 20 - 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. $$ \begin{array}{l}{x-3 y \geq-7} \\ {5 x+y \leq 13} \\ {x+6 y \geq-9} \\ {3 x-2 y \geq-7} \\ {f(x, y)=x-y}\end{array} $$

Step-by-Step Solution

Answer

Plot inequalities, find vertices, evaluate \(f(x, y)\) at vertices to find max. and min values.

Step 1: Graph Each Inequality

First, graph each inequality. 1. For the inequality \(x - 3y \geq -7\), rearrange it to \(y \leq \frac{x+7}{3}\) and draw the line \(y = \frac{x+7}{3}\). Shade the region below the line since it's a 'greater than or equal to' inequality. 2. For \(5x + y \leq 13\), rearrange as \(y \leq -5x + 13\), plot the line \(y = -5x + 13\), and shade below.3. For \(x + 6y \geq -9\), rearrange to \(y \geq -\frac{x+9}{6}\), plot \(y = -\frac{x+9}{6}\), and shade above.4. For \(3x - 2y \geq -7\), rearrange as \(y \leq \frac{3x+7}{2}\), plot \(y = \frac{3x+7}{2}\), and shade below.

Step 2: Identify the Feasible Region

The feasible region is where all the shaded areas from the inequalities overlap. Use a graph to visualize where all conditions are met simultaneously. This region is bounded by the lines of the system of inequalities.

Step 3: Find the Vertices of the Feasible Region

Find the points of intersection of the lines forming the boundary of the feasible region, solving the system of equations pairwise:1. \(x - 3y = -7\) and \(5x + y = 13\)2. \(x - 3y = -7\) and \(x + 6y = -9\)3. \(5x + y = 13\) and \(x + 6y = -9\)4. \(5x + y = 13\) and \(3x - 2y = -7\)5. \(x + 6y = -9\) and \(3x - 2y = -7\)6. \(x - 3y = -7\) and \(3x - 2y = -7\)Calculate the intersections to identify the vertices.

Step 4: Evaluate the Function at Each Vertex

Substitute the coordinates of each vertex into the function \(f(x, y) = x - y\). Calculate the function's value at each vertex to determine which values are maximum and minimum.

Step 5: Determine the Maximum and Minimum Values

Compare the calculated values from the previous step to identify the maximum and minimum values. These correspond to the maximum and minimum values of the function within the feasible region.

Key Concepts

Graphing Systems of Inequalities

A system of inequalities is a set of two or more inequalities with the same variables. Graphing each inequality helps in visually understanding the solutions for the system.

To graph an inequality, follow these simple steps:

• Write the inequality in slope-intercept form, which is usually in the format of \( y = mx + b\).
• Plot the corresponding line on the graph using its slope \( m \) and y-intercept \( b \).
• If the inequality is 'greater than or equal to' (\( \geq \)) or 'less than or equal to' (\( \leq \)), draw a solid line. Use a dashed line for strict inequalities (\( > \) or \( < \)).
• Shade the region above or below the line, depending on the inequality sign.

By graphing each inequality of the system, you can identify the area where all of the inequalities are simultaneously satisfied. This is crucial for solving further parts of the problem.

Feasible Region

The feasible region is a central concept when graphing systems of inequalities. This region represents all the possible solutions that satisfy all conditions of the inequalities at the same time.

After graphing each inequality, the next step is finding where these shaded regions overlap. The overlapping area is known as the feasible region:

• This region is always bounded in two-dimensional space by the lines from the given inequalities.
• Every point within this region is a potential solution to the system.
• To clearly identify this region, look for intersections of the boundary lines that form polygons.

Understanding the feasible region helps in finding solutions that meet all specified constraints, allowing for practical applications in fields like economics, engineering, and logistics.

Systems of Equations

In systems of inequalities, the concept of systems of equations emerges when solving for intersection points. These points are where inequality boundaries meet, forming the vertices of the feasible region.

To find these intersection points:

• Take equations of the boundary lines (converted from inequalities) and solve them pairwise as a system of equations.
• Use methods like substitution or elimination to calculate the exact coordinates of these points.
• Each pair of equations will yield one point of intersection if the lines intersect within the graph's range.

Finding these intersection points enables us to derive all possible boundary solutions of the system and aids in assessing the performance of various functions over this region.

Function Optimization

Function optimization involves finding the maximum or minimum values of a given function within a specific region. When working with systems of inequalities, this often means evaluating the function at the vertices of the feasible region.

Here's how to optimize a function in this context:

• Substitute each vertex's coordinates into the function.
• Compute the function's value for each point to see how high or low it reaches.
• Compare these values to determine which one is the greatest (maximum) and which is the smallest (minimum).

Knowing how to optimize a function is essential for making efficient decisions, like maximizing profits or minimizing costs, within constrained environments.