Final answer:
To determine the value of c for which the points (5, 10), (6, 7), and (c, 7) lie on a line, we calculate the slope of the line passing through the first two points. This slope is found to be -3. Using the slope and the coordinates of the second point, we calculate that c must be 6, which is not an option provided in the question.
Step-by-step explanation:
To find the value of c such that the points (5, 10), (6, 7), and (c, 7) lie on a line, we need to determine the slope of the line passing through the first two points and use that to find the corresponding x-value for the third point where the y-value is 7. The slope m of a line passing through two points (x1, y1) and (x2, y2) is calculated as:
m = (y2 - y1) / (x2 - x1)
For points (5, 10) and (6, 7), the slope is:
m = (7 - 10) / (6 - 5) = -3 / 1 = -3
Since the point (c, 7) must have the same slope, we can use the point-slope form of the line equation (y - y1) = m (x - x1), with (x1, y1) = (6, 7) and m = -3:
7 - 7 = -3 (c - 6)
0 = -3c + 18
3c = 18
c = 18 / 3
c = 6
However, this value of c is not one of the provided options, which suggests there may be an error in the question or in the provided options. If one of the options must be chosen, none of them are correct. Therefore, we cannot select a valid option from A, B, C, and D.