Math Problem Statement
graph the lines y=-4x+3 and y=x-7 and find the point of intersection
Solution
numpy as np import matplotlib.pyplot as plt
Define the equations of the lines
x = np.linspace(-10, 10, 400) y1 = -4 * x + 3 # Equation: y = -4x + 3 y2 = x - 7 # Equation: y = x - 7
Find the point of intersection by solving the equations
-4x + 3 = x - 7
-5x = -10
x = 2
x_intersection = 2 y_intersection = -4 * x_intersection + 3 # Substitute x into one of the equations
Plot the lines
plt.figure(figsize=(8, 6)) plt.plot(x, y1, label='y = -4x + 3', color='blue') plt.plot(x, y2, label='y = x - 7', color='orange') plt.scatter(x_intersection, y_intersection, color='red', label=f'Intersection ({x_intersection}, {y_intersection})', zorder=5)
Add labels, legend, and grid
plt.title('Graph of the Lines and their Intersection') plt.xlabel('x-axis') plt.ylabel('y-axis') plt.axhline(0, color='black',linewidth=0.5, linestyle='--') plt.axvline(0, color='black',linewidth=0.5, linestyle='--') plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5) plt.legend() plt.show()
x_intersection, y_intersectionThe graph of the lines and is shown above. The lines intersect at the point .
Would you like a detailed explanation of how the intersection point was calculated or any other details?
Related Questions
- How would the graph change if the slope of one line is altered?
- Can you confirm the intersection using substitution or elimination methods algebraically?
- What is the slope of a line perpendicular to ?
- How does the y-intercept affect the positioning of a line on the graph?
- What happens if we add a third line, , to the system?
Tip: When graphing lines, always check their slopes to understand how they will intersect or run parallel.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Equations
Graphing
Intersection of Lines
Formulas
y = mx + b (Slope-Intercept Form)
Equation solving to find intersection points
Theorems
Concept of the intersection of two lines
Suitable Grade Level
Grades 8-10
Related Recommendation
Graphing Lines y=-4x+3 and y=x-7 and Finding the Point of Intersection
Graphing Two Intersecting Lines: y = 2x + 3 and y = -x + 1
Solve Systems of Linear Equations Graphically: y = -x - 4 and x - 2y = -4
Solving the System of Equations y = -2x - 7 and x - y = -2 Graphically
Solving System of Linear Equations Graphically: y = 4x - 2 and y = -x + 3