Final answer:
To find the value of k when the points (1,4), (6,-3), and (16,k) lie on a straight line, we can calculate the slope between (1,4) and (6,-3). Then, we can set up an equation using the slope between (6,-3) and (16,k) and solve for k. The value of k is -17.
Step-by-step explanation:
To determine the value of k when the points (1,4), (6,-3), and (16,k) lie on a straight line, we can use the concept of slope. The slope between two points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)
Let's calculate the slope between the points (1,4) and (6,-3):
slope = (-3 - 4) / (6 - 1) = -7/5
Since the points (1,4), (6,-3), and (16,k) must lie on the same line, the slope between the points (6,-3) and (16,k) should also be -7/5. We can use this information to find the value of k:
-7/5 = (k - (-3)) / (16 - 6)
After simplifying, we get:
-7/5 = (k + 3) / 10
Cross multiplying:
-70 = 5k + 15
Subtracting 15 from both sides:
-85 = 5k
Dividing both sides by 5:
k = -17