a) Points: $A(3, -7)$ and $B(4, 8)$

1. Slope (m):

The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For $A(3, -7)$ and $B(4, 8)$:

$m = \frac{8 - (-7)}{4 - 3} = \frac{8 + 7}{1} = \frac{15}{1} = 15$

2. Inclination (θ):

The inclination of a line is the angle $θ$ it makes with the positive direction of the x-axis. It can be found using:

$\theta = \tan^{-1}(m)$

$\theta = \tan^{-1}(15)$

Using a calculator, the inclination is approximately:

$\theta \approx 86.2^\circ$

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b) Points: $A(9, 0)$ and $B(3, 18)$

1. Slope (m):

Using the same formula for slope:

$m = \frac{18 - 0}{3 - 9} = \frac{18}{-6} = -3$

2. Inclination (θ):

Now, find the inclination:

$\theta = \tan^{-1}(-3)$

Using a calculator:

$\theta \approx -71.57^\circ$

However, inclination is usually given as a positive angle, so:

$\theta = 180^\circ + (-71.57^\circ) = 108.43^\circ$

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Final Answer:

• a) Slope = 15, Inclination $\theta \approx 86.2^\circ$
• b) Slope = -3, Inclination $\theta \approx 108.43^\circ$

Do you need more details or have any questions?

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Relative Questions:

1. How is the slope affected by vertical or horizontal lines?
2. What is the slope of a line that passes through two points with the same x-coordinate?
3. How do you calculate the slope and inclination of parallel lines?
4. How does the inclination change if the slope is negative?
5. What would the inclination be for a line with a slope of 0?

Tip: The inclination is always measured counterclockwise from the positive x-axis.