A System Has The Following Constraints:${ \begin{align} x + Y & \geq 80 \ x & \geq 0 \ 3x + 2y & \leq 360 \ y & \geq 0 \ x + 2y & \leq 200 \ \end{align} }$Which Graph Represents The Feasible Region For The System?

Mar 8, 2025 by ADMIN 217 views

A System of Linear Inequalities: Finding the Feasible Region

Introduction

In this article, we will explore a system of linear inequalities and determine the feasible region that satisfies all the given constraints. The system consists of five linear inequalities, and we will use graphical methods to visualize the feasible region. We will also discuss the importance of the feasible region in optimization problems and how it can be used to find the optimal solution.

Understanding the System of Linear Inequalities

The system of linear inequalities is given by:

\begin{align*} x + y & \geq 80 \\ x & \geq 0 \\ 3x + 2y & \leq 360 \\ y & \geq 0 \\ x + 2y & \leq 200 \\ \end{align*}

These inequalities represent the constraints of the system, and we need to find the region that satisfies all of them. The first inequality, $x + y \geq 80$ , represents the constraint that the sum of $x$ and $y$ must be greater than or equal to 80. The second inequality, $x \geq 0$ , represents the constraint that $x$ must be greater than or equal to 0. The third inequality, $3x + 2y \leq 360$ , represents the constraint that the sum of 3 times $x$ and 2 times $y$ must be less than or equal to 360. The fourth inequality, $y \geq 0$ , represents the constraint that $y$ must be greater than or equal to 0. The fifth inequality, $x + 2y \leq 200$ , represents the constraint that the sum of $x$ and 2 times $y$ must be less than or equal to 200.

Graphing the Inequalities

To visualize the feasible region, we need to graph the inequalities on a coordinate plane. We will start by graphing the first inequality, $x + y \geq 80$ . This inequality represents a line with a slope of -1 and a y-intercept of 80. We will graph the line and shade the region above it to represent the constraint.

### Graph of the First Inequality

Next, we will graph the second inequality, $x \geq 0$ . This inequality represents the y-axis, and we will shade the region to the right of the y-axis to represent the constraint.

### Graph of the Second Inequality

Now, we will graph the third inequality, $3x + 2y \leq 360$ . This inequality represents a line with a slope of -3/2 and a y-intercept of 180. We will graph the line and shade the region below it to represent the constraint.

### Graph of the Third Inequality

Next, we will graph the fourth inequality, $y \geq 0$ . This inequality represents the x-axis, and we will shade the region above the x-axis to represent the constraint.

### Graph of the Fourth Inequality

Finally, we will graph the fifth inequality, $x + 2y \leq 200$ . This inequality represents a line with a slope of -1/2 and a y-intercept of 100. We will graph the line and shade the region below it to represent the constraint.

### Graph of the Fifth Inequality

Finding the Feasible Region

To find the feasible region, we need to find the intersection of all the shaded regions. The feasible region is the region that satisfies all the constraints, and it is the area where all the shaded regions overlap.

### Feasible Region

The feasible region is a polygon with vertices at (0,80), (120,0), (40,60), and (0,0). This region represents the set of all possible solutions that satisfy all the constraints.

Conclusion

In this article, we have explored a system of linear inequalities and determined the feasible region that satisfies all the given constraints. We have used graphical methods to visualize the feasible region and have discussed the importance of the feasible region in optimization problems. The feasible region is a polygon with vertices at (0,80), (120,0), (40,60), and (0,0), and it represents the set of all possible solutions that satisfy all the constraints.

Importance of the Feasible Region

The feasible region is an important concept in optimization problems, as it represents the set of all possible solutions that satisfy all the constraints. The feasible region can be used to find the optimal solution by identifying the vertex of the polygon that maximizes or minimizes the objective function.

Applications of the Feasible Region

The feasible region has many applications in real-world problems, such as:

Resource allocation: The feasible region can be used to allocate resources in a way that satisfies all the constraints.
Scheduling: The feasible region can be used to schedule tasks in a way that satisfies all the constraints.
Optimization: The feasible region can be used to find the optimal solution by identifying the vertex of the polygon that maximizes or minimizes the objective function.

Final Thoughts

In conclusion, the feasible region is an important concept in optimization problems, and it represents the set of all possible solutions that satisfy all the constraints. The feasible region can be used to find the optimal solution by identifying the vertex of the polygon that maximizes or minimizes the objective function. The feasible region has many applications in real-world problems, such as resource allocation, scheduling, and optimization.

Introduction

In our previous article, we explored a system of linear inequalities and determined the feasible region that satisfies all the given constraints. In this article, we will answer some frequently asked questions about the system of linear inequalities and the feasible region.

Q: What is the feasible region?

A: The feasible region is the set of all possible solutions that satisfy all the constraints of the system of linear inequalities. It is the area where all the shaded regions overlap.

Q: How do I find the feasible region?

A: To find the feasible region, you need to graph the inequalities on a coordinate plane and find the intersection of all the shaded regions. The feasible region is the area where all the shaded regions overlap.

Q: What are the vertices of the feasible region?

A: The vertices of the feasible region are the points where the lines intersect. In this case, the vertices are (0,80), (120,0), (40,60), and (0,0).

Q: How do I use the feasible region to find the optimal solution?

A: To find the optimal solution, you need to identify the vertex of the polygon that maximizes or minimizes the objective function. The feasible region can be used to find the optimal solution by identifying the vertex of the polygon that maximizes or minimizes the objective function.

Q: What are some applications of the feasible region?

A: The feasible region has many applications in real-world problems, such as:

Resource allocation: The feasible region can be used to allocate resources in a way that satisfies all the constraints.
Scheduling: The feasible region can be used to schedule tasks in a way that satisfies all the constraints.
Optimization: The feasible region can be used to find the optimal solution by identifying the vertex of the polygon that maximizes or minimizes the objective function.

Q: How do I graph the inequalities on a coordinate plane?

A: To graph the inequalities on a coordinate plane, you need to:

Plot the lines on the coordinate plane.
Shade the region above the line for the inequality $x + y \geq 80$ .
Shade the region to the right of the y-axis for the inequality $x \geq 0$ .
Shade the region below the line for the inequality $3x + 2y \leq 360$ .
Shade the region above the x-axis for the inequality $y \geq 0$ .
Shade the region below the line for the inequality $x + 2y \leq 200$ .

Q: What are some common mistakes to avoid when graphing the inequalities?

A: Some common mistakes to avoid when graphing the inequalities include:

Not plotting the lines on the coordinate plane: Make sure to plot the lines on the coordinate plane before shading the regions.
Not shading the correct region: Make sure to shade the correct region for each inequality.
Not considering the constraints: Make sure to consider the constraints of the system of linear inequalities when graphing the inequalities.

Q: How do I use the feasible region to solve optimization problems?

A: To use the feasible region to solve optimization problems, you need to:

Identify the objective function.
Graph the inequalities on a coordinate plane.
Find the feasible region.
Identify the vertex of the polygon that maximizes or minimizes the objective function.
Use the vertex to find the optimal solution.

Q: What are some real-world applications of the feasible region?

A: The feasible region has many real-world applications, such as:

Resource allocation: The feasible region can be used to allocate resources in a way that satisfies all the constraints.
Scheduling: The feasible region can be used to schedule tasks in a way that satisfies all the constraints.
Optimization: The feasible region can be used to find the optimal solution by identifying the vertex of the polygon that maximizes or minimizes the objective function.

Conclusion

In this article, we have answered some frequently asked questions about the system of linear inequalities and the feasible region. We have discussed how to find the feasible region, how to use the feasible region to find the optimal solution, and some real-world applications of the feasible region.