To show that the points $A(3, 9)$, $B(-2, -16)$, and $C\left(\frac{1}{5}, -5\right)$ are collinear, we need to verify that they lie on the same straight line. One way to check this is by ensuring that the slope between any two pairs of points is the same.

Step 1: Find the slope between points A and B

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

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

Using points $A(3, 9)$ and $B(-2, -16)$, the slope $m_{AB}$ is:

$m_{AB} = \frac{-16 - 9}{-2 - 3} = \frac{-25}{-5} = 5$

Step 2: Find the slope between points B and C

Next, calculate the slope between $B(-2, -16)$ and $C\left(\frac{1}{5}, -5\right)$:

$m_{BC} = \frac{-5 - (-16)}{\frac{1}{5} - (-2)} = \frac{-5 + 16}{\frac{1}{5} + 2} = \frac{11}{\frac{1}{5} + \frac{10}{5}} = \frac{11}{\frac{11}{5}} = 5$

Step 3: Find the slope between points A and C

Finally, calculate the slope between $A(3, 9)$ and $C\left(\frac{1}{5}, -5\right)$:

$m_{AC} = \frac{-5 - 9}{\frac{1}{5} - 3} = \frac{-14}{\frac{1}{5} - \frac{15}{5}} = \frac{-14}{\frac{-14}{5}} = 5$

Conclusion:

Since the slopes $m_{AB} = m_{BC} = m_{AC} = 5$, the points $A$, $B$, and $C$ are collinear, meaning they lie on the same straight line.

Let me know if you want more details or have any questions!

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Here are 5 related questions for further exploration:

1. How can we find the equation of the line passing through the three points?
2. What does it mean geometrically when three points are collinear?
3. How can you use vectors to show collinearity of points?
4. Can the concept of slope be applied to vertical lines?
5. How do we check if four points are coplanar in three dimensions?

Tip: When verifying collinearity, ensuring equal slopes between all pairs of points is a reliable method!