To plot the intercepts and graph the equation $3x - 2y = -6$, we need to find the x-intercept and y-intercept.

Finding the Intercepts

X-intercept

The x-intercept occurs when $y = 0$.

$3x - 2(0) = -6 \implies 3x = -6 \implies x = -2$

So, the x-intercept is $(-2, 0)$.

Y-intercept

The y-intercept occurs when $x = 0$.

$3(0) - 2y = -6 \implies -2y = -6 \implies y = 3$

So, the y-intercept is $(0, 3)$.

Graphing the Equation

1. Plot the x-intercept $(-2, 0)$.
2. Plot the y-intercept $(0, 3)$.
3. Draw a straight line through these two points.

Here's the graph of the equation:

x & y & \text{Points} \\ \hline -2 & 0 & (-2, 0) \\ 0 & 3 & (0, 3) \\ \end{array}$$ Now, let's plot this. ```python import matplotlib.pyplot as plt import numpy as np # Define the function to plot the line def plot_line(): x = np.linspace(-4, 4, 400) y = (3/2)*x + 3 plt.figure(figsize=(8, 6)) plt.plot(x, y, '-r', label='3x - 2y = -6') plt.plot(-2, 0, 'bo') # x-intercept plt.plot(0, 3, 'bo') # y-intercept plt.axhline(0, color='black',linewidth=0.5) plt.axvline(0, color='black',linewidth=0.5) plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5) plt.legend(loc='best') plt.title('Graph of 3x - 2y = -6') plt.xlabel('x') plt.ylabel('y') plt.xlim(-4, 4) plt.ylim(-4, 4) plt.show() plot_line() ``` The graph shows the line passing through the intercepts $$(-2, 0)$$ and $$(0, 3)$$. Would you like more details or have any questions? Here are 5 more questions you might want to explore: 1. How do you find the slope of a line given an equation? 2. How do you convert a standard form equation to slope-intercept form? 3. How do you find the equation of a line given two points? 4. What are parallel and perpendicular lines in terms of slopes? 5. How can you determine if a point lies on a given line? **Tip:** When graphing linear equations, always start by finding the intercepts to make the graphing process easier.