Introduction:
To determine the value of k, we need to find the condition for the three given points (-2,5),(-5,-10), and (k,-13) to be collinear. Collinear points lie on the same line, meaning that the slope between any two points is the same.

Calculating the slope:
To find the slope between two points (x1,y1) and (x2,y2), we use the formula:
slope = (y2 - y1) / (x2 - x1)

Using this formula, we can calculate the slope between the first two given points:
slope1 = (-10 - 5) / (-5 - (-2))
= -15 / (-3)
= 5

Equating slopes:
Since the three points are collinear, the slope between the first and third point should also be 5. Substituting the values into the slope formula, we have:
slope2 = (-13 - 5) / (k - (-2))
= -18 / (k + 2)

Now, we can equate the two slopes and solve for k:
slope1 = slope2
5 = -18 / (k + 2)

Solving for k:
To eliminate the fraction, we can cross multiply:
5 * (k + 2) = -18

Expanding the equation:
5k + 10 = -18

Next, we isolate the variable by subtracting 10 from both sides:
5k = -18 - 10
5k = -28

Finally, we solve for k by dividing both sides by 5:
k = -28 / 5
k = -5.6

Conclusion:
The value of k that makes the points (-2,5),(-5,-10), and (k,-13) collinear is k = -5.6. This means that when k is equal to -5.6, the three points lie on the same line.

K = -28/5