
Slope of a line
A line contains the points (3,8) and (4,13). Find three more points that lie on the line.
(I want to solve this without a graphing calculator)
I know the answer is (a), but how do you figure it out?
a) (2,3), (5,8), (6,23)
b) (2,3), (5,23), (6,18)
c) (2,3), (5,18), (6,23)
d) (2,23), (5,18), (6,3)

Since you titled this "slope of the line", I presume you know how to find that: use the two given points to find the slope of the line. Then find the slope of the line containing one of the given points and one of the possible "answer" points. If the two slopes are not the same, move on to the next answer. If the two slopes are the same, you will need to check other points in that same answer.
However, you are wrong that "a" is the answer a line through (5, 8) and (3, 8) has slope 0. Perhaps you have copied something wrong none of those answers is correct.

The fastest way no calcs is to plot the given points. It is then easy to see which pairs are on the line if any.

None of these are the answer
Taking the gradient joining (3,8) and (4,13) which is 5
Then creating a linear equation from it: $\displaystyle y=5x7 $
Now sub points in to see if both sides equal:
Sub (2,3) $\displaystyle 3=5(2)7$ > $\displaystyle 3=107 $ which is correct
subbing (5,8) in: $\displaystyle 8 =5(5) 7 $ this can't be right. So it can't be option a)
Subbing in (5,23) from option b) doesn't work either
Subbing in (2,3) from option c) doesn't work...
Finally sub in (2,23) from option d)... doesn't work
I don't think there's an answer in here...

a) (2,3), (5,8), (6,23)
these aren't even in the same line
the slope of (2,3), (5,8)is =5/3
but the slope of(5,8), (6,23) is=15

What if there were a TYPO?
(5, 18)
That was hard to see...