Let's go step by step for each pair of points. To calculate the slope of a line through two points, we use the formula:

$\text{slope} (m) = \frac{y_2 - y_1}{x_2 - x_1}$

Where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

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(i) Points: (5, 3) and (4, 5)

1. Substitute the points into the slope formula:

$m = \frac{5 - 3}{4 - 5} = \frac{2}{-1} = -2$

So, the slope of the line is -2.

2. Graphing the line:
   • Plot the points (5, 3) and (4, 5).
   • Draw the line through them.

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(ii) Points: (-2, -3) and (4, 5)

1. Substitute the points into the slope formula:

$m = \frac{5 - (-3)}{4 - (-2)} = \frac{5 + 3}{4 + 2} = \frac{8}{6} = \frac{4}{3}$

So, the slope of the line is 4/3.

2. Graphing the line:
   • Plot the points (-2, -3) and (4, 5).
   • Draw the line through them.

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(iii) Points: (5, 3) and (7, 3)

1. Substitute the points into the slope formula:

$m = \frac{3 - 3}{7 - 5} = \frac{0}{2} = 0$

So, the slope of the line is 0. This means the line is horizontal.

2. Graphing the line:
   • Plot the points (5, 3) and (7, 3).
   • Draw a horizontal line passing through these points.

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Now, I'll generate the graphs for these three sets of points and their respective lines. Let me plot them for you. matplotlib.pyplot as plt

Coordinates for the points

points_i = [(5, 3), (4, 5)] points_ii = [(-2, -3), (4, 5)] points_iii = [(5, 3), (7, 3)]

Create the plot

fig, ax = plt.subplots(1, 3, figsize=(18, 6))

Plot (i) points (5, 3) and (4, 5)

ax[0].plot([5, 4], [3, 5], marker='o', color='blue', label="Line (i)") ax[0].set_title("Line for (5, 3) and (4, 5)\nSlope = -2") ax[0].set_xlabel("x") ax[0].set_ylabel("y") ax[0].grid(True) ax[0].set_xlim(3, 6) ax[0].set_ylim(2, 6) ax[0].legend()

Plot (ii) points (-2, -3) and (4, 5)

ax[1].plot([-2, 4], [-3, 5], marker='o', color='green', label="Line (ii)") ax[1].set_title("Line for (-2, -3) and (4, 5)\nSlope = 4/3") ax[1].set_xlabel("x") ax[1].set_ylabel("y") ax[1].grid(True) ax[1].set_xlim(-3, 5) ax[1].set_ylim(-4, 6) ax[1].legend()

Plot (iii) points (5, 3) and (7, 3)

ax[2].plot([5, 7], [3, 3], marker='o', color='red', label="Line (iii)") ax[2].set_title("Line for (5, 3) and (7, 3)\nSlope = 0") ax[2].set_xlabel("x") ax[2].set_ylabel("y") ax[2].grid(True) ax[2].set_xlim(4, 8) ax[2].set_ylim(2, 4) ax[2].legend()

plt.tight_layout() plt.show()Here are the graphs for each of the given pairs of points:

• (i) Line through (5, 3) and (4, 5): Slope is -2, and the line slopes downwards.
• (ii) Line through (-2, -3) and (4, 5): Slope is 4/3, a positive slope that rises as you move from left to right.
• (iii) Line through (5, 3) and (7, 3): Slope is 0, indicating a horizontal line.

Do you need further details on the calculations or graphs? Let me know if you'd like any modifications!

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Here are some related questions you might find interesting:

1. How does the slope affect the angle of a line?
2. How do we find the equation of a line when we know the slope and one point?
3. What happens if two points have the same x-coordinate?
4. How do you determine if two lines are parallel or perpendicular?
5. Can a vertical line have a slope? If so, what is it?

Tip: When calculating slopes, remember that a vertical line has an undefined slope, while a horizontal line has a slope of 0.