BANON wrote:
Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p ?
(1) Lines n and p intersect at the point (5 , 1).
(2) The y-intercept of line n is greater than the y-intercept of line p.
We need to determine whether the slope of line n is less than the slope of line p.
Statement One Alone:
Lines n and p intersect at the point (5,1).
If the two lines intersect at a point, they are not parallel and hence their slopes are not equal (unless they are identical lines). So the slope of one line must be greater than the slope of the other line. However, we can’t determine which line has the greater slope. Statement one alone is not sufficient. Eliminate answer choices A and D.
Statement Two Alone:
The y-intercept of line n is greater than the y-intercept of line p.
Knowing that the y-intercept of one line is greater than the y-intercept of the other does not allow us to determine which line has the greater slope.
For example, line n could have a y-intercept 2 and slope 3, and line p could have a y-intercept 1 and slope 2. In this case, line p has the lesser slope. However, it’s also possible that line n could have a y-intercept 2 and slope 2, and line p could have a y-intercept 1 and slope 3. In which case, line n has the lesser slope. Statement two alone is not sufficient. Eliminate answer choice B.
Statements One and Two Together:
Knowing the point where the two lines intersect and the relationship of the y-intercept of each line allows us to determine which line has the lesser slope.
Even though we don’t know the actual y-intercept of each line, we know that the y-intercept of line n is greater than that of line p. So we can let the y-intercept of line n be b, and that of line p be c where b > c.
Thus, line n passes through (0, b), and line p passes through (0, c). Both lines also pass through (5, 1). Let’s calculate their slopes:
Slope of line n = (1 – b)/(5 – 0) = (1 – b)/5
Slope of line p = (1 – c)/(5 – 0) = (1 – c)/5
Now let’s determine whether (1 – b)/5 < (1 – c)/5.
Is (1 – b)/5 < (1 – c)/5 ?
Is 1 – b < 1 – c ?
Is –b < –c ?
Is b > c ?
Since, from the information in statement two, we know that b is greater than c, we have answered the question: the slope of line n is indeed less than that of line p.
Answer: C
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