AbdurRakib wrote:
Line \(\ell\) lies in the xy-plane and does not pass through the origin. What is the slope of line \(\ell\) ?
(1) The x-intercept of line \(\ell\) is twice the y-intercept of line \(\ell\)
(2) The x-and y-intercepts of line \(\ell\) are both positive
Target question: What is the slope of line l? Statement 1: The x-intercept of line \(\ell\) is twice the y-intercept of line l Let k = the y-intercept of line l
This means 2k = the x-intercept of line l
If the y-intercept is k, then line l passes through the y-axis at the point
(0, k)If the x-intercept is 2k, then line l passes through the x-axis at the point
(2k, 0)Since
(0, k) and
(2k, 0) are both points on line l, we can apply the slope formula to these points to find the slope of line l.
We get: slope = (k - 0)/(0 - 2k) = k/(-2k) = -1/2
So,
the slope of line l = -1/2Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: The x-and y-intercepts of line l are both positive If we're able to imagine different lines (with DIFFERENT SLOPES) that satisfy this condition, we'll quickly see that statement 2 is not sufficient. However, if we don't automatically see this, we can take the following approach...
There are many different cases that satisfy statement 2 yet yield different answers to the target question. Here are two:
Case a: the x-intercept is 1 and the y-intercept is 1, which means line l passes through (1, 0) and (0, 1). Applying the slope formula, we get:
slope = (0 - 1)/(1 - 0) = -1Case b: the x-intercept is 2 and the y-intercept is 1, which means line l passes through (2, 0) and (0, 1). Applying the slope formula, we get:
slope = (0 - 1)/(2 - 0) = -1/2Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer: A