Graph the equation \( y=-\frac{3}{2} x+3 \)

To graph the linear equation \( y = -\frac{3}{2}x + 3 \), follow these steps: ### 1. Identify Key Components of the Equation The given equation is in **slope-intercept form**, which is: \[ y = mx + b \] where: - \( m \) is the **slope** of the line. - \( b \) is the **y-intercept** (the point where the line crosses the y-axis). For the equation \( y = -\frac{3}{2}x + 3 \): - **Slope (\( m \))**: \( -\frac{3}{2} \) - **Y-intercept (\( b \))**: \( 3 \) ### 2. Plot the Y-Intercept Start by plotting the y-intercept on the graph: - **Point**: \( (0, 3) \) This is where the line crosses the y-axis. ### 3. Use the Slope to Find Another Point The slope \( m = -\frac{3}{2} \) indicates that for every **2 units** you move **horizontally** (in the x-direction), you move **3 units downward** (since the slope is negative). - **From (0, 3)**: - Move **2 units right** to \( x = 2 \). - Move **3 units down** to \( y = 0 \). - **Second Point**: \( (2, 0) \) ### 4. Draw the Line Using the two points \( (0, 3) \) and \( (2, 0) \), draw a straight line that extends in both directions. This line represents all solutions to the equation \( y = -\frac{3}{2}x + 3 \). ### 5. (Optional) Find the X-Intercept The x-intercept is where the line crosses the x-axis (\( y = 0 \)). Set \( y = 0 \) in the equation and solve for \( x \): \[ 0 = -\frac{3}{2}x + 3 \\ \frac{3}{2}x = 3 \\ x = 2 \] - **X-Intercept**: \( (2, 0) \) ### 6. Summary of Key Points - **Y-Intercept**: \( (0, 3) \) - **X-Intercept**: \( (2, 0) \) - **Slope**: \( -\frac{3}{2} \) (downward slope) ### 7. Visual Representation While I can't display an actual graph here, you can visualize the line as follows: 1. **Start at (0, 3)** on the y-axis. 2. **Move to (2, 0)** on the x-axis using the slope. 3. **Draw a straight line** through these points extending infinitely in both directions. ### 8. Sample Graph Using Python If you're interested in plotting the graph yourself using Python, here's a simple script using the `matplotlib` library: ```python import matplotlib.pyplot as plt import numpy as np # Define the equation y = (-3/2)x + 3 def y(x): return (-3/2) * x + 3 # Generate x values x = np.linspace(-4, 4, 400) y_vals = y(x) # Create the plot plt.figure(figsize=(8,6)) plt.plot(x, y_vals, label=r'$y = -\frac{3}{2}x + 3$') # Plot intercepts plt.plot(0, 3, 'ro') # y-intercept plt.plot(2, 0, 'bo') # x-intercept plt.text(0, 3, ' (0,3)', verticalalignment='bottom', horizontalalignment='right') plt.text(2, 0, ' (2,0)', verticalalignment='top', horizontalalignment='left') # Add grid, labels, and title plt.grid(True) plt.xlabel('x') plt.ylabel('y') plt.title('Graph of y = -3/2 x + 3') plt.legend() plt.axhline(0, color='black', linewidth=0.5) # x-axis plt.axvline(0, color='black', linewidth=0.5) # y-axis # Show the plot plt.show() ``` **Instructions to Run the Script:** 1. Ensure you have Python installed on your computer. 2. Install the `matplotlib` and `numpy` libraries if you haven't already by running: ``` pip install matplotlib numpy ``` 3. Save the script to a file, say `graph.py`. 4. Run the script using the command: ``` python graph.py ``` This will display the graph of the equation with the intercepts marked. ### 9. Conclusion The equation \( y = -\frac{3}{2}x + 3 \) represents a straight line with a negative slope, indicating that as \( x \) increases, \( y \) decreases. The y-intercept is at \( (0, 3) \), and the x-intercept is at \( (2, 0) \). Using these points, you can easily graph the line on a coordinate plane.