import numpy as np import matplotlib.pyplot as plt

Given point and slope

point = (4, 6) m = -5 / 2

Generate x values

x = np.linspace(0, 10, 100)

Equation of the line: y - y1 = m(x - x1)

y = m(x - x1) + y1

y = m * (x - point[0]) + point[1]

Plot the line

plt.figure(figsize=(8, 6)) plt.plot(x, y, label=f"y = {m}(x - {point[0]}) + {point[1]}") plt.scatter(*point, color='red', label=f"Point ({point[0]}, {point[1]})")

Adding labels, title, and legend

plt.title("Graph of the Line Through (4, 6) with Slope -5/2") plt.xlabel("x") plt.ylabel("y") plt.axhline(0, color='black', linewidth=0.8, linestyle='--') plt.axvline(0, color='black', linewidth=0.8, linestyle='--') plt.legend() plt.grid()

Display the plot

plt.show() <<ImageDisplayed>>successHere is the graph of the line passing through the point (4, 6) with a slope of -5/2. The line's equation is derived as $y = -\frac{5}{2}(x - 4) + 6$.

Do you want further details or have any specific questions? Here are five related questions you might consider:

1. How is the slope of the line related to its steepness and direction?
2. Can we find another point on this line to verify the equation?
3. How do we determine the x- and y-intercepts of this line?
4. What happens to the line if we change the slope or the point it passes through?
5. How do we graph a line if given in standard form, like $Ax + By = C$?

Tip: To find the intercepts quickly, set $x = 0$ for the y-intercept and $y = 0$ for the x-intercept!