gmatt1476
In the rectangular coordinate system, line k passes through the point (n, −1). Is the slope of line k greater than zero?
(1) Line k passes through the origin.
(2) Line k passes through the point (1, n + 2).
DS35210.01
Given: Line k passes through the point (n, −1) Target question: Is the slope of line k greater than zero?Key concept: If points \((a, b)\) and \((c, d)\) both lie on a line, then the slope of that line \(= \frac{d-b}{c-a}\) Statement 1: Line k passes through the origin. In other words, line k passes through the point \((0,0)\)
Since we also know line k passes through the point \((n, −1)\), the slope of line k \(= \frac{(-1)-0}{n-0}=\frac{-1}{n}\)
So, if \(n = 1\), the slope of line k \(= \frac{-1}{1} = -1\), which means the answer to the target question is
NO.Conversely, if \(n = -1\), the slope of line k \(= \frac{-1}{-1} = 1\), which means the answer to the target question is
YES.Since we can’t answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: Line k passes through the point (1, n + 2).Since we also know line k passes through the point \((n, −1)\), the slope of line k \(= \frac{(n+2)-(-1)}{1-n}=\frac{n+3}{1-n}\)
So, if \(n = 2\), the slope of line k \(= \frac{2+3}{1-2} = -5\), which means the answer to the target question is
NO.Conversely, if \(n = 0\), the slope of line k \(= \frac{0+3}{1-0} = 3\), which means the answer to the target question is
YES.Since we can’t answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined Key concept: The slope between any two points on a line will always be the same. The following three points all lie on line k: (n, −1), (0, 0) and (1, n + 2)
This means that slope between (n, −1) and (0, 0) = the slope between (0, 0) and (1, n + 2)
In other words: \(\frac{n - 0}{(-1) - 0} = \frac{(n+2)-0}{1-0}\)
Simplify: \(\frac{n}{-1} = \frac{n+2}{1}\)
Cross multiply: (-1)(n+2) = (n)(1)
Expand: -n - 2 = n
Solve:
n = -1Now that we know the value of n, we COULD perform all of the calculations necessary to determine the slope of line k (and thus answer the target question).
Since we could answer the target question with certainty, the combined statements are sufficient.
Answer: C