To solve this problem, we need to graph the two linear equations on a coordinate system based on their slope-intercept form $y = mx + b$, where:

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

Given Equations:

1. $y = \frac{2}{5}x - 2$

   • Slope ($m$) = $\frac{2}{5}$
   • Y-intercept ($b$) = $-2$

2. $y = -\frac{3}{2}x + 4$

   • Slope ($m$) = $-\frac{3}{2}$
   • Y-intercept ($b$) = $4$

Steps to Graph:

1. For Equation (a):

   • Plot the y-intercept $(-2)$ on the y-axis.
   • Use the slope $m = \frac{2}{5}$ to determine the rise over run:
     • From the point $(0, -2)$, go up 2 units (rise) and right 5 units (run).
   • Plot another point and draw a straight line through these two points.

2. For Equation (b):

   • Plot the y-intercept $4$ on the y-axis.
   • Use the slope $m = -\frac{3}{2}$ (negative slope):
     • From $(0, 4)$, go down 3 units (rise) and right 2 units (run).
   • Plot another point and draw a straight line through these two points.

I will now create and display the graph.import matplotlib.pyplot as plt import numpy as np

Define the equations

x = np.linspace(-10, 10, 400)

First equation: y = (2/5)x - 2

y1 = (2/5) * x - 2

Second equation: y = -(3/2)x + 4

y2 = -(3/2) * x + 4

Plot the lines

plt.figure(figsize=(8, 6))

Plot first equation

plt.plot(x, y1, label=r"$y = \frac{2}{5}x - 2$", color="blue")

Plot second equation

plt.plot(x, y2, label=r"$y = -\frac{3}{2}x + 4$", color="red")

Add axes

plt.axhline(0, color='black',linewidth=0.8) plt.axvline(0, color='black',linewidth=0.8)

Add grid, labels, legend

plt.grid(color='gray', linestyle='--', linewidth=0.5) plt.title("Graph of Linear Functions") plt.xlabel("x-axis") plt.ylabel("y-axis") plt.legend() plt.show() <<ImageDisplayed>>successHere is the graph of the two linear functions:

1. The blue line represents $y = \frac{2}{5}x - 2$.
2. The red line represents $y = -\frac{3}{2}x + 4$.

These lines are drawn based on their slopes and y-intercepts.

Let me know if you have any questions or need further details. Here are some related questions to deepen your understanding:

1. How can you determine the slope from a linear equation in slope-intercept form?
2. What happens to a line's graph if you change the slope to a steeper value?
3. How can you find the intersection point of two linear equations algebraically?
4. What is the significance of the y-intercept in a graph, and how can you use it?
5. How would the graph change if you multiplied the slope of each equation by $-1$?

Tip: When graphing, always verify points by substituting into the equation to ensure accuracy!