nalinnair wrote:
In the xy-plane, lines k and l intersect at the point (1,1). Is the y-intercept of k greater than the y-intercept of l?
(1) The slope of k is less than the slope of l.
(2) The slope of l is positive.
Solution:
We are told that lines k and l both contain the point (1, 1). The question is whether the y-intercept of k is greater than the y-intercept of l. Let a be the y-intercept of line k and b be the y-intercept of line l. The question now becomes whether a > b.
Statement One Alone:The slope of k is less than the slope of l
Since the y-intercept of line k is a, line k passes through (0, a). We also know that line k passes through (1, 1), so we can apply the slope formula with these two points:
Slope k = (a - 1)/(0 - 1) = (a - 1)/(-1) = 1 - a
Similarly, since the y-intercept of line l is b, line l passes through (0, b). Let’s apply the slope formula using the points (0, b) and (1, 1):
Slope l = (b - 1)/(0 - 1) = 1 - b
Now, statement 1 tells us that slope of k is less than slope of l. Thus:
1 - a < 1 - b
-a < -b
a > b
Recall that a was the y-intercept of line k and b was the y-intercept of line l. It follows that the y-intercept of line k is greater than the y-intercept of line l. This answers the question. Statement one alone is sufficient.
Statement Two Alone:The slope of l is positive
Recall that in the analysis of statement one, we actually showed that if two lines pass through the point (1, 1), then the line with the smaller slope has greater y-intercept. In one scenario, assume that the slope of line k is 1 and the slope of line l is 2. In this case, the y-intercept of line k is greater than the y-intercept of line l. On the other hand, if the slope of line k is 2 and the slope of line l is 1, then the y-intercept of line k is smaller than the y-intercept of line l. Notice that both of these scenarios satisfy “the slope of line l is positive.” Since we have multiple possible answers to the question, statement two alone is not sufficient to answer the question.
Answer: A