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In the xy-plane, line k passes through the point (2, 3). What is the slope of line k?
(1) Line k crosses the x-axis at a point whose x-coordinate is positive.
(2) The y-intercept of line k is positive.
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Correct answer
E
Write line k as y = mx + b through (2, 3), so 3 = 2m + b and b = 3 − 2m. The slope is m, and the question asks whether the statements pin m.
Statement (1): the x-intercept is at x = −b/m = (2m − 3)/m, and this is positive when m > 1.5 or when m < 0. That is a wide range, so the slope is not determined. For instance m = 2 (x-intercept at 0.5) and m = −1 (x-intercept at 5) both have a positive x-intercept. Insufficient.
Statement (2): the y-intercept is b = 3 − 2m, positive when m < 1.5. Many slopes qualify (m = 1 and m = −1 both give a positive y-intercept), so insufficient.
Together: m must satisfy (m > 1.5 or m < 0) AND m < 1.5, which leaves exactly m < 0. That is still a whole family of negative slopes (m = −1 and m = −4 both work), so the slope is not pinned. Neither statement implies the other (m = 2 meets only the first, m = 1 meets only the second). Answer: (E).
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