Let's first rewrite the given linear equation:

$2x + 3y = 3$

To graph it, we can convert this equation into slope-intercept form ($y = mx + b$), where $m$ is the slope and $b$ is the y-intercept.

Step 1: Solve for $y$

3y = -2x + 3 \quad \text{(Subtract 2x from both sides)} y = -\frac{2}{3}x + 1 \quad \text{(Divide by 3 to isolate y)}

Step 2: Identify the slope and y-intercept

• The slope $m = -\frac{2}{3}$
• The y-intercept $b = 1$

Step 3: Plot the graph

1. Start at the y-intercept: Place a point at $(0, 1)$ on the y-axis.
2. Use the slope: From the point $(0, 1)$, move down 2 units (because of the negative slope) and right 3 units to get the next point at $(3, -1)$.
3. Draw the line: Connect the points, and extend the line in both directions.

I will generate the graph now for you! matplotlib.pyplot as plt import numpy as np

Define the equation y = -2/3 * x + 1

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

Plot the graph

plt.figure(figsize=(6,6)) plt.plot(x, y, label=r'$2x + 3y = 3$', color='blue')

Mark the y-intercept

plt.scatter(0, 1, color='red', zorder=5) # y-intercept (0,1)

Annotate the y-intercept

plt.text(0, 1, '(0, 1)', fontsize=12, ha='right', color='red')

Label the axes and title

plt.axhline(0, color='black',linewidth=1) plt.axvline(0, color='black',linewidth=1) plt.title('Graph of the Linear Equation: $2x + 3y = 3$') plt.xlabel('x') plt.ylabel('y')

Set the limits of the graph

plt.xlim(-10, 10) plt.ylim(-10, 10)

Show the grid

plt.grid(True)

Display the plot

plt.legend() plt.show()Here is the graph of the linear equation $2x + 3y = 3$. The line crosses the y-axis at $(0, 1)$, which is the y-intercept. The slope of the line is $-\frac{2}{3}$, meaning for every 3 units you move to the right, the line moves 2 units down.

Do you need any further details or have any questions about the graph?

Here are 5 related questions for further exploration:

1. How do you calculate the slope of a line from two points?
2. What does it mean geometrically when the slope of a line is negative?
3. How can you find the x-intercept of a linear equation?
4. How do you graph a linear equation in standard form, like $Ax + By = C$?
5. What are the characteristics of parallel and perpendicular lines?

Tip: When graphing linear equations, always identify the slope and y-intercept first to quickly plot points and draw the line!