To graph the equation $3y = -6 + x$ using its slope and $y$-intercept, let's first rewrite it in the slope-intercept form, $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Steps:

1. Start with the given equation: $3y = -6 + x$

2. Rearrange to isolate $y$: $y = \frac{x}{3} - 2$

   Here, the slope $m = \frac{1}{3}$ and the $y$-intercept $b = -2$.

Graphing:

1. Plot the $y$-intercept $(0, -2)$ on the graph.
2. Use the slope $\frac{1}{3}$ (rise over run). From $(0, -2)$:
   • Move up 1 unit (rise).
   • Move right 3 units (run) to locate the next point $(3, -1)$.
3. Draw a straight line passing through these points.

Let me visualize this for you. matplotlib.pyplot as plt import numpy as np

Define the equation y = (1/3)x - 2

x = np.linspace(-10, 10, 400) y = (1/3) * x - 2

Create the plot

plt.figure(figsize=(8, 6)) plt.axhline(0, color='black',linewidth=0.5) plt.axvline(0, color='black',linewidth=0.5) plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5)

Plot the line

plt.plot(x, y, label=r"$y = \frac{1}{3}x - 2$", color="blue")

Plot the points: y-intercept and another point using the slope

plt.scatter(0, -2, color="red", label="y-intercept (0, -2)") plt.scatter(3, -1, color="green", label="Point (3, -1)")

Add labels and legend

plt.title("Graph of y = (1/3)x - 2") plt.xlabel("x") plt.ylabel("y") plt.legend() plt.show()Here is the graph of the equation $y = \frac{1}{3}x - 2$:

• The red point $(0, -2)$ represents the $y$-intercept.
• The green point $(3, -1)$ is another point on the line, determined using the slope $\frac{1}{3}$.

Would you like a step-by-step breakdown of the calculations, or do you have any questions?

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Related Questions:

1. How do you find the slope and $y$-intercept from an equation in standard form?
2. What is the significance of the slope in a graph?
3. How do you convert between slope-intercept and standard form of a linear equation?
4. How do you graph a line given two points instead of slope and intercept?
5. Can a slope of 0 or undefined appear in equations, and what do their graphs look like?

Tip:

When working with slopes, always think of the fraction as "rise over run" to make plotting easier!