Suppose That The Following System Of Inequalities Represents The Constraints In A Linear Programming Application. Graph The Feasible Region On Paper, Then Answer The Questions That Follow.$\[ \begin{aligned} 6x - 6y & \leq 18 \\ -3x + 9y & \leq 27

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Understanding the System of Inequalities

In linear programming, a system of inequalities is used to represent the constraints of a problem. The constraints are used to define the feasible region, which is the set of all possible solutions to the problem. In this article, we will graph the feasible region of a given system of inequalities and answer the questions that follow.

The System of Inequalities

The system of inequalities is given by:

6x−6y≤18−3x+9y≤27\begin{aligned} 6x - 6y & \leq 18 \\ -3x + 9y & \leq 27 \end{aligned}

Graphing the Inequalities

To graph the feasible region, we need to graph the two inequalities separately and then find the intersection of the two regions.

Graphing the First Inequality

The first inequality is 6x−6y≤186x - 6y \leq 18. To graph this inequality, we can rewrite it as y≥x−3y \geq x - 3. This is a linear inequality in two variables, and it can be graphed as a line with a shaded region above it.

# Graph of the First Inequality

## Line: y = x - 3

## Shaded Region: y ≥ x - 3

Graphing the Second Inequality

The second inequality is −3x+9y≤27-3x + 9y \leq 27. To graph this inequality, we can rewrite it as y≥13x−3y \geq \frac{1}{3}x - 3. This is also a linear inequality in two variables, and it can be graphed as a line with a shaded region above it.

# Graph of the Second Inequality

## Line: y = (1/3)x - 3

## Shaded Region: y ≥ (1/3)x - 3

Finding the Intersection of the Two Regions

To find the intersection of the two regions, we need to find the points where the two lines intersect. We can do this by setting the two equations equal to each other and solving for x.

# Finding the Intersection of the Two Regions

## Setting the Two Equations Equal to Each Other

x - 3 = (1/3)x - 3

## Solving for x

(2/3)x = 0

x = 0

## Finding the Corresponding y-Value

y = x - 3
y = 0 - 3
y = -3

## The Intersection Point is (0, -3)

Graphing the Feasible Region

The feasible region is the intersection of the two shaded regions. It is a polygon with vertices at (0, -3), (6, 3), and (0, 3).

# Graph of the Feasible Region

## Polygon with Vertices at (0, -3), (6, 3), and (0, 3)

Answering the Questions

Now that we have graphed the feasible region, we can answer the questions that follow.

Question 1: What is the feasible region of the given system of inequalities?

The feasible region is a polygon with vertices at (0, -3), (6, 3), and (0, 3).

Question 2: What is the intersection of the two shaded regions?

The intersection of the two shaded regions is a polygon with vertices at (0, -3), (6, 3), and (0, 3).

Question 3: What is the value of x at the intersection point?

The value of x at the intersection point is 0.

Question 4: What is the value of y at the intersection point?

The value of y at the intersection point is -3.

Conclusion

In this article, we graphed the feasible region of a given system of inequalities and answered the questions that follow. The feasible region is a polygon with vertices at (0, -3), (6, 3), and (0, 3). The intersection of the two shaded regions is also a polygon with vertices at (0, -3), (6, 3), and (0, 3). The value of x at the intersection point is 0, and the value of y at the intersection point is -3.

Understanding the System of Inequalities

In linear programming, a system of inequalities is used to represent the constraints of a problem. The constraints are used to define the feasible region, which is the set of all possible solutions to the problem. In this article, we will answer some frequently asked questions about graphing the feasible region of a given system of inequalities.

Q&A

Q1: What is the feasible region of the given system of inequalities?

A1: The feasible region is a polygon with vertices at (0, -3), (6, 3), and (0, 3).

Q2: How do I graph the first inequality?

A2: To graph the first inequality, you can rewrite it as y≥x−3y \geq x - 3. This is a linear inequality in two variables, and it can be graphed as a line with a shaded region above it.

Q3: How do I graph the second inequality?

A3: To graph the second inequality, you can rewrite it as y≥13x−3y \geq \frac{1}{3}x - 3. This is also a linear inequality in two variables, and it can be graphed as a line with a shaded region above it.

Q4: How do I find the intersection of the two regions?

A4: To find the intersection of the two regions, you need to find the points where the two lines intersect. You can do this by setting the two equations equal to each other and solving for x.

Q5: What is the value of x at the intersection point?

A5: The value of x at the intersection point is 0.

Q6: What is the value of y at the intersection point?

A6: The value of y at the intersection point is -3.

Q7: How do I graph the feasible region?

A7: The feasible region is the intersection of the two shaded regions. It is a polygon with vertices at (0, -3), (6, 3), and (0, 3).

Q8: What are the vertices of the feasible region?

A8: The vertices of the feasible region are (0, -3), (6, 3), and (0, 3).

Q9: How do I determine the feasible region of a system of inequalities?

A9: To determine the feasible region of a system of inequalities, you need to graph the inequalities and find the intersection of the two regions.

Q10: What is the importance of the feasible region in linear programming?

A10: The feasible region is the set of all possible solutions to the problem. It is the region where the constraints of the problem are satisfied.

Conclusion

In this article, we answered some frequently asked questions about graphing the feasible region of a given system of inequalities. We discussed the importance of the feasible region in linear programming and provided step-by-step instructions on how to graph the feasible region.

Additional Resources

  • Linear Programming Tutorial: A comprehensive tutorial on linear programming, including the basics of linear programming, the simplex method, and more.
  • Graphing Inequalities: A tutorial on graphing inequalities, including linear inequalities and quadratic inequalities.
  • Feasible Region: A tutorial on the feasible region, including how to determine the feasible region of a system of inequalities.

Final Thoughts

Graphing the feasible region of a system of inequalities is an important step in linear programming. By understanding the feasible region, you can determine the set of all possible solutions to the problem. We hope this article has been helpful in answering your questions about graphing the feasible region.