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 y=4x+3y = -4x + 3 and y=x7y = x - 7 is shown above. The lines intersect at the point (2,5)(2, -5).

Would you like a detailed explanation of how the intersection point was calculated or any other details?

Related Questions

  1. How would the graph change if the slope of one line is altered?
  2. Can you confirm the intersection using substitution or elimination methods algebraically?
  3. What is the slope of a line perpendicular to y=4x+3y = -4x + 3?
  4. How does the y-intercept affect the positioning of a line on the graph?
  5. What happens if we add a third line, y=2x+1y = 2x + 1, 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