Geometry

[kamalakarthi]

Hi, I would like to know if there is better approach to this question instead of solving for equations. Can you please help.

[Ian Stewart]

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?

[Brent@GMATPrepNow]

In an xy-coordinate plane, a line is defined by y = kx + 1. If (4, b), (a, 4), and (a, b +1) are three points on the line, where a and b are unknown, then k = ?

a) 1/2
b) 1
c) 1 1/2
d) 2
e) 2 1/2

There's a HUGE CLUE in the fact that (a, 4) and (a, b+1) are both on the same line. Notice that the x-coordinates are the same. If the x-coordinates are the same, then there are two possible scenarios:
scenario #1: The points (a, 4) and (a, b+1) are DIFFERENT points, in which case the line is vertical (with undefined slope)
scenario #2: The points (a, 4) and (a, b+1) define the SAME point

IMPORTANT: If a line is defined by y = kx + 1, then k represents the slope. So, the question is really asking us to find the slope of the line.

In scenario #1, the slope would be undefined. Since none of the answer choices are undefined, we can rule out scenario #1, which means (a, 4) and (a, b+1) define the SAME point. So, we can be certain that b+1 = 4, which means b = 3

Now that we know that b = 3, we can use the fact that the point (4,b) is on the line.
This means that the point (4,3) is on the line y = kx + 1.
When we plug x=4 and y=3 into the equation, we get 3 = (k)(4) + 1
Solve to get k = 1/2

Answer: A

Cheers,
Brent

[Matt@VeritasPrep]

Equations are the best way here, but we could always go for another set of them from the ones given above.

k = the slope of the line, and since k = 0 isn't an option in the answers, know the line ISN'T horizontal. We also know that k is defined, so the line can't be perfectly vertical either.

Since the line isn't horizontal or vertical, we can't have two points with the same x or y-coordinate on the line. (a, 4) and (a, b + 1) have the same x-coordinate, and we just saw that they can't be different points, so they must be the SAME POINT, and 4 = b + 1.

That gives b = 3, from which we know that (4, 3) and (a, 4) are two points on the line.

Plugging (4, 3) into the equation y = kx + 1 gives us 3 = 4k + 1, or k = 1/2.