Math Problem Statement
Make a grap of a system of linear inequality in two variables. (Make a grap or detailed illustration that same with the picture shown
Solution
To recreate the graph of the system of linear inequalities shown in the images, let me analyze the equations and graph their solution regions systematically.
Equations and Analysis from the Images:
-
Equation 1:
- x-intercept: (when )
- y-intercept: (when )
-
Equation 2:
- x-intercept: (when )
- y-intercept: (when )
Steps for Recreating the Graph:
- Graph each inequality, shading the correct solution regions.
- Mark the boundary lines as dashed or solid based on the inequalities:
- Dashed for or .
- Solid for or .
I will now generate a graph similar to the one in your provided image. matplotlib.pyplot as plt import numpy as np
Define the range of x-values for the graph
x = np.linspace(-2, 10, 500)
Inequalities (solving for y)
y1 = 1 - x # From x + y <= 1 y2 = 2 - 2 * x # From 2x + y >= 2
Set up the plot
plt.figure(figsize=(10, 8))
Plot the boundary lines
plt.plot(x, y1, label=r'$x + y = 1$', color='blue', linestyle='--') plt.plot(x, y2, label=r'$2x + y = 2$', color='green', linestyle='--')
Fill the solution regions
plt.fill_between(x, y1, -10, where=(y1 >= -10), color='blue', alpha=0.3, label='Region for $x + y \leq 1$') plt.fill_between(x, y2, 10, where=(y2 <= 10), color='green', alpha=0.3, label='Region for $2x + y \geq 2$')
Axis settings
plt.axhline(0, color='black', linewidth=0.8) plt.axvline(0, color='black', linewidth=0.8) plt.xlim(-2, 10) plt.ylim(-10, 10)
Add labels, legend, and grid
plt.xlabel("x-axis") plt.ylabel("y-axis") plt.title("Graph of the System of Linear Inequalities") plt.legend() plt.grid(alpha=0.5) plt.show()Here is the graph of the system of linear inequalities and . The blue shaded region represents the solution for , while the green shaded region represents the solution for . The overlap of the two regions (darker area) indicates the combined solution to the system of inequalities.
Do you want further clarification or adjustments?
Here are 5 related questions for further exploration:
- How do we determine the feasibility region for a system of inequalities?
- How would the graph change if the inequalities were strict ( or )?
- What are the vertices of the overlapping region, and how can we calculate them algebraically?
- Can this system of inequalities represent a real-world scenario? If so, provide an example.
- How can we verify that a specific point satisfies both inequalities?
Tip: Always verify the solution region by testing a point within the shaded area to confirm it satisfies all given inequalities.
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Math Problem Analysis
Mathematical Concepts
Linear Inequalities
Graphing in Two Variables
Coordinate Geometry
Formulas
x-intercept formula: Let y = 0 to solve for x
y-intercept formula: Let x = 0 to solve for y
Shading solution regions based on inequality signs
Theorems
Linear inequality graphing rules
Solution regions for inequalities
Suitable Grade Level
Grade 8
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