If (a, 4) and (a, b+1) are on the same line, and that line isn't vertical, they must be the same point, because they have the same x-coordinate. So b+1 = 4, and b = 3. We know (4, b) is on the line, so if b = 3, the point (4, 3) is on the line. That means the point must work in the line's equation:

y = kx + 1

Plugging in x = 4 and y = 3, we have

3 = 4k + 1
k = 1/2

There's a way to do this without using as much algebra, but you need to understand slopes well. If a line has this equation:

y = kx + 1

then its y-intercept is 1, so the point (0, 1) is on the line. When we find that (4, 3) is also on the line, we know that when we move across 4 units, from x=0 to x=4, the line rises by 2 units, from y=1 to y=3. Since slope = rise/run, slope = 2/4 = 1/2. Since k is the slope of the line, that's the answer. Or once you have the two points, you could just use the slope formula.

All that said, the answer choices are bizarre. A GMAT question would never include "1 and 1/2" as an answer choice to a slope question; they would always write "5/2". What is the source?