Identify The Graph Of The System Of Inequalities:${ \begin{aligned} y & \geq 2x \ 10x + 20y & \leq 300 \end{aligned} }$

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Introduction

Graphing systems of inequalities is a fundamental concept in mathematics, particularly in algebra and geometry. It involves representing a set of inequalities as a graph on a coordinate plane. In this article, we will focus on identifying the graph of the system of inequalities:

y≥2x10x+20y≤300\begin{aligned} y & \geq 2x \\ 10x + 20y & \leq 300 \end{aligned}

Understanding the Inequalities

Before we dive into graphing the system of inequalities, let's understand the individual inequalities.

Inequality 1: y ≥ 2x

This inequality represents a line with a slope of 2 and a y-intercept of 0. The symbol ≥ indicates that the line is included in the graph, and all points above and on the line satisfy the inequality.

Inequality 2: 10x + 20y ≤ 300

This inequality represents a line with a slope of -1/2 and a y-intercept of 15. The symbol ≤ indicates that the line is included in the graph, and all points below and on the line satisfy the inequality.

Graphing the Inequalities

To graph the system of inequalities, we need to graph each inequality separately and then find the intersection of the two graphs.

Graphing Inequality 1: y ≥ 2x

To graph this inequality, we can use the slope-intercept form of a line (y = mx + b), where m is the slope and b is the y-intercept. In this case, the slope is 2 and the y-intercept is 0.

import matplotlib.pyplot as plt
import numpy as np

m = 2
b = 0



x = np.linspace(-10, 10, 400)



y = m * x + b



plt.plot(x, y, label='y ≥ 2x')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of y ≥ 2x')
plt.legend()
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

This code will generate a graph of the line y ≥ 2x.

Graphing Inequality 2: 10x + 20y ≤ 300

To graph this inequality, we can use the slope-intercept form of a line (y = mx + b), where m is the slope and b is the y-intercept. In this case, the slope is -1/2 and the y-intercept is 15.

import matplotlib.pyplot as plt
import numpy as np


m = -1/2
b = 15



x = np.linspace(-10, 10, 400)



y = m * x + b



plt.plot(x, y, label='10x + 20y ≤ 300')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of 10x + 20y ≤ 300')
plt.legend()
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

This code will generate a graph of the line 10x + 20y ≤ 300.

Finding the Intersection

To find the intersection of the two graphs, we need to solve the system of equations:

y=2x10x+20y=300\begin{aligned} y & = 2x \\ 10x + 20y & = 300 \end{aligned}

We can substitute the first equation into the second equation to get:

10x+20(2x)=30010x+40x=30050x=300x=6\begin{aligned} 10x + 20(2x) & = 300 \\ 10x + 40x & = 300 \\ 50x & = 300 \\ x & = 6 \end{aligned}

Substituting x = 6 into the first equation, we get:

y=2(6)y=12\begin{aligned} y & = 2(6) \\ y & = 12 \end{aligned}

Therefore, the intersection of the two graphs is the point (6, 12).

Graphing the System of Inequalities

To graph the system of inequalities, we need to graph each inequality separately and then find the intersection of the two graphs.

import matplotlib.pyplot as plt
import numpy as np


m1 = 2
b1 = 0
m2 = -1/2
b2 = 15



x = np.linspace(-10, 10, 400)



y1 = m1 * x + b1
y2 = m2 * x + b2



plt.plot(x, y1, label='y ≥ 2x')
plt.plot(x, y2, label='10x + 20y ≤ 300')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of the System of Inequalities')
plt.legend()
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

This code will generate a graph of the system of inequalities.

Conclusion

Q: What is the difference between a system of equations and a system of inequalities?

A: A system of equations is a set of equations that are solved simultaneously to find the values of the variables. A system of inequalities, on the other hand, is a set of inequalities that are solved simultaneously to find the values of the variables.

Q: How do I graph a system of inequalities?

A: To graph a system of inequalities, you need to graph each inequality separately and then find the intersection of the two graphs. You can use the slope-intercept form of a line (y = mx + b) to graph each inequality.

Q: What is the intersection of two graphs?

A: The intersection of two graphs is the point where the two graphs meet. To find the intersection of two graphs, you need to solve the system of equations formed by the two inequalities.

Q: How do I find the intersection of two graphs?

A: To find the intersection of two graphs, you need to solve the system of equations formed by the two inequalities. You can use substitution or elimination to solve the system of equations.

Q: What is the difference between a solid line and a dashed line on a graph?

A: A solid line on a graph represents an equality, while a dashed line represents an inequality. A solid line is used to graph an equation, while a dashed line is used to graph an inequality.

Q: How do I determine which line to use when graphing an inequality?

A: When graphing an inequality, you need to use a dashed line if the inequality symbol is ≥ or ≤, and a solid line if the inequality symbol is > or <.

Q: Can I graph a system of inequalities with more than two inequalities?

A: Yes, you can graph a system of inequalities with more than two inequalities. However, it may be more difficult to find the intersection of the graphs.

Q: How do I graph a system of inequalities with fractions?

A: To graph a system of inequalities with fractions, you need to simplify the fractions and then graph the inequalities as usual.

Q: Can I use a graphing calculator to graph a system of inequalities?

A: Yes, you can use a graphing calculator to graph a system of inequalities. However, you need to make sure that the calculator is set to graph inequalities correctly.

Q: How do I check my work when graphing a system of inequalities?

A: To check your work when graphing a system of inequalities, you need to make sure that the graphs are correct and that the intersection of the graphs is correct.

Q: What are some common mistakes to avoid when graphing a system of inequalities?

A: Some common mistakes to avoid when graphing a system of inequalities include:

  • Graphing the wrong inequality
  • Using the wrong line type (solid or dashed)
  • Not finding the intersection of the graphs
  • Not checking the work

Q: How do I use graphing to solve real-world problems?

A: Graphing can be used to solve a wide range of real-world problems, including optimization problems, linear programming problems, and systems of equations. By graphing the inequalities and finding the intersection of the graphs, you can find the solution to the problem.

Q: What are some applications of graphing systems of inequalities?

A: Some applications of graphing systems of inequalities include:

  • Optimization problems
  • Linear programming problems
  • Systems of equations
  • Real-world problems

Q: How do I use technology to graph a system of inequalities?

A: You can use a graphing calculator or a computer program to graph a system of inequalities. These tools can help you to graph the inequalities and find the intersection of the graphs.

Q: What are some tips for graphing a system of inequalities?

A: Some tips for graphing a system of inequalities include:

  • Make sure to graph each inequality separately
  • Use the correct line type (solid or dashed)
  • Find the intersection of the graphs
  • Check the work
  • Use technology to help with graphing and finding the intersection of the graphs.