Let me extract the mathematical problem from the image and provide a solution.The problem in the image asks to:

1. Write the equation of a line in slope-intercept form with the given slope $\frac{3}{2}$ and $y$-intercept $-1$.
2. Graph the line.

Step 1: Write the Equation in Slope-Intercept Form

The slope-intercept form of a line is: $y = mx + b$ where:

• $m$ is the slope
• $b$ is the $y$-intercept.

Here, $m = \frac{3}{2}$ and $b = -1$. Substituting these values, the equation becomes: $y = \frac{3}{2}x - 1$

Step 2: Graph the Line

I will plot the graph for the line $y = \frac{3}{2}x - 1$. One moment while I generate it. numpy as np import matplotlib.pyplot as plt

Define the equation of the line

def line_eq(x): return (3/2) * x - 1

Generate x values

x = np.linspace(-5, 5, 100) y = line_eq(x)

Plot the line

plt.figure(figsize=(8, 6)) plt.plot(x, y, label=r"$y = \frac{3}{2}x - 1$", color='blue') plt.axhline(0, color='black', linewidth=0.8, linestyle='--') # x-axis plt.axvline(0, color='black', linewidth=0.8, linestyle='--') # y-axis

Highlight the y-intercept and a point from slope calculation

plt.scatter([0], [-1], color='red', label="y-intercept (-1)") # y-intercept plt.scatter([2], [2], color='green', label="Point from slope (2, 2)") # Using slope to mark (2, 2)

Labels and legend

plt.title("Graph of the Line $y = \frac{3}{2}x - 1$") plt.xlabel("x") plt.ylabel("y") plt.grid(color='gray', linestyle='--', linewidth=0.5) plt.legend() plt.show()### Final Answer: The equation of the line is: $y = \frac{3}{2}x - 1$

The graph above represents the line. It includes:

• The $y$-intercept at $(0, -1)$.
• Another point on the line at $(2, 2)$, calculated using the slope $\frac{3}{2}$.

Do you have any questions or need more details?

Related Questions:

1. How is the slope calculated from two points on a line?
2. How do we verify if a given point lies on this line?
3. What happens to the graph if the slope is negative?
4. How would you write the equation if the slope and $y$-intercept were fractions?
5. Can you explain how to convert this equation to standard form?

Tip: For consistent graphing, always mark the intercepts and use the slope to determine additional points!