In the rectangular coordinate system, line k passes through

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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).




OA C

Source: Official Guide

[Scott@TargetTestPrep]

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).




OA C

Source: Official Guide

Solution:

We need to determine whether the slope of line k is positive.

Statement One Only:
Line k passes through the origin.

Since line k passes through the origin i.e., (0, 0), we can calculate the slope of line k as:

(-1 - 0)/(n - 0) = -1/n

However, since we don’t know the value of n, we can’t determine whether the slope of line k is positive. For example, if n is negative, then -1/n is positive. However, if n is positive, then -1/n is negative.

Statement Two Only:
Line k passes through the point (1, n + 2).

Since line k passes through the point (1, n + 2), we can calculate the slope of line k as:

(-1 - (n + 2))/(n - 1) = (-n - 3)/(n - 1) = (n + 3)/(1 - n)

However, since we don’t know the value of n, we can’t determine whether the slope of line k is positive. For example, if n is 0, then (n + 3)/(1 - n) = 3 is positive. However, if n is 2, then (n + 3)/(1 - n) = -5 is negative.

Statements One and Two Together:

From statement one, we see that the slope of line k is -1/n and from statement two, we see that the slope is (n + 3)/(1 - n). Since the slope of any line must be unique, we have:

-1/n = (n + 3)/(1 - n)

-1(1 - n) = n(n + 3)

n - 1 = n^2 + 3n

n^2 + 2n + 1 = 0

(n + 1)^2 = 0

n + 1 = 0

n = -1

Since n = -1 and the slope of line k is -1/n, the slope is -1/(-1) = 1. We see that the slope of line k is positive. Statements one and two together are sufficient to answer the question.

Answer: C