The problem in the image states:

The points P(4, -6), Q(12, 2k), and R(13, 4) lie on a straight line. Find the possible values of k.

Since the points are collinear (lie on a straight line), the slopes between any two pairs of points must be equal. We can calculate the slope between points $P$ and $Q$, and between points $Q$ and $R$, and set them equal to each other.

Step 1: Slope between $P(4, -6)$ and $Q(12, 2k)$

The slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substitute the coordinates of $P(4, -6)$ and $Q(12, 2k)$: $m_{PQ} = \frac{2k - (-6)}{12 - 4} = \frac{2k + 6}{8} = \frac{k + 3}{4}$

Step 2: Slope between $Q(12, 2k)$ and $R(13, 4)$

Substitute the coordinates of $Q(12, 2k)$ and $R(13, 4)$: $m_{QR} = \frac{4 - 2k}{13 - 12} = 4 - 2k$

Step 3: Set the slopes equal

Since the points are collinear, the slopes must be equal: $\frac{k + 3}{4} = 4 - 2k$

Step 4: Solve for $k$

Multiply both sides by 4 to eliminate the denominator: $k + 3 = 16 - 8k$ Move all terms involving $k$ to one side: $k + 8k = 16 - 3$ $9k = 13$ $k = \frac{13}{9}$

Thus, the value of $k$ is $\frac{13}{9}$.

Would you like more details or have any questions about this solution?

Here are some related questions you might find helpful:

1. How do you derive the slope formula for two points?
2. What is the significance of the collinearity condition in coordinate geometry?
3. Can you explain how to check if three points form a triangle instead of being collinear?
4. How do different values of $k$ affect the position of the point $Q$?
5. What happens if two or more points in a set are identical in terms of slope?

Tip: Whenever working with collinear points, equating slopes is a powerful and direct way to solve for unknowns in the coordinates.