swerve wrote:In the xy-plane, the line k passes through the origin and through the point (a,b), where ab does not equal 0. Is b positive?
(1) The slope of line k is negative
(2) a < b
Target question: Is b (the y-coordinate of the point on the line) positive?
Given: Line k passes through the origin and through the point (a,b)
Statement 1: The slope of line k is negative
There are several lines and points that satisfy statement 1. Here are two:
Case a:
In this case, b (y-coordinate) is positive. So, the answer to the target question is
YES, b is positive
Case b:
In this case, b (y-coordinate) is negative. So, the answer to the target question is
NO, b is NOT positive
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: a < b
There are several lines and points that satisfy statement 2. Here are two:
Case a:
In this case, b (y-coordinate) is positive. So, the answer to the target question is
YES, b is positive
Case b:
In this case, b (y-coordinate) is negative. So, the answer to the target question is
NO, b is NOT positive
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
Statement 1 tells us that the slope of line k is negative. This means line k passes through quadrants II and IV.
In quadrant II, a (the x-coordinate) is always negative, and b (the y-coordinate) is always positive
In quadrant IV, a (the x-coordinate) is always positive, and b (the y-coordinate) is always negative
Statement 2 tells us that a < b
This means that the point (a,b) must be in quadrant II
(because, all points in quadrant IV are such that the x-coordinate (a) is greater than the y-coordinate (b)
If point (a,b) is in quadrant II, we can be certain that
b (the y-coordinate) is positive
Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
