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Geometry

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kamalakarthi Senior | Next Rank: 100 Posts Default Avatar
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Geometry

Post Mon Jul 24, 2017 6:49 pm
Hi, I would like to know if there is better approach to this question instead of solving for equations. Can you please help.
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Post Sun Aug 06, 2017 10:41 pm
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.

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Post Tue Jul 25, 2017 6:10 am
Quote:
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

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Brent

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Post Sun Aug 06, 2017 10:41 pm
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.

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