题目信息
In the standard (x,y) coordinate plane, what is the slope of the line containing the distinct points P and Q ?
A:Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B:Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C:BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D:EACH statement ALONE is sufficient.
E:Statements (1) and (2) TOGETHER are NOT sufficient.
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正确答案：
E:Statements (1) and (2) TOGETHER are NOT sufficient.
Geometry Simple coordinate geometry
If P = (1,0) and Q = (0,1), then both P and Q lie on the graph of |x| + |y| = 1 and the slope of the line containing P and Q is −1. However, if P = (1,0) and Q = (−1,0), then both P and Q lie on the graph of |x| + |y| = 1 and the slope of the line containing P and Q is 0; NOT sufficient. If P = (1,0) and Q = (0,1), then both P and Q lie on the graph of |x + y| = 1 and the slope of the line containing P and Q is −1. However, if P = (1,0) and Q = (−1,0), then both P and Q lie on the graph of |x + y| = 1 and the slope of the line containing P and Q is 0; NOT sufficient.
Taking (1) and (2) together is still not sufficient because the same examples used in (1) were also used in (2).
Although it is not necessary to visualize the graphs of |x| + |y| = 1 and |x + y| = 1 to solve this problem, some readers may be interested in their graphs. The graph of |x| + |y| = 1 is a square with vertices at the four points (±1,0) and (0,±1). This can be seen by graphing x + y = 1 in the first quadrant, which gives a line segment with endpoints (1,0) and (0,1), and then reflecting this line segment about one or both coordinate axes for the other quadrants (e.g., in the second quadrant, x < 0 and y > 0, and so |x| + |y| = 1 becomes −x + y = 1). The graph of |x + y| = 1 is the union of two lines, one with equation x + y = 1 and the other with equation x + y = −1.
The correct answer is E;both statements together are still not sufficient.
If P = (1,0) and Q = (0,1), then both P and Q lie on the graph of |x| + |y| = 1 and the slope of the line containing P and Q is −1. However, if P = (1,0) and Q = (−1,0), then both P and Q lie on the graph of |x| + |y| = 1 and the slope of the line containing P and Q is 0; NOT sufficient. If P = (1,0) and Q = (0,1), then both P and Q lie on the graph of |x + y| = 1 and the slope of the line containing P and Q is −1. However, if P = (1,0) and Q = (−1,0), then both P and Q lie on the graph of |x + y| = 1 and the slope of the line containing P and Q is 0; NOT sufficient.
Taking (1) and (2) together is still not sufficient because the same examples used in (1) were also used in (2).
Although it is not necessary to visualize the graphs of |x| + |y| = 1 and |x + y| = 1 to solve this problem, some readers may be interested in their graphs. The graph of |x| + |y| = 1 is a square with vertices at the four points (±1,0) and (0,±1). This can be seen by graphing x + y = 1 in the first quadrant, which gives a line segment with endpoints (1,0) and (0,1), and then reflecting this line segment about one or both coordinate axes for the other quadrants (e.g., in the second quadrant, x < 0 and y > 0, and so |x| + |y| = 1 becomes −x + y = 1). The graph of |x + y| = 1 is the union of two lines, one with equation x + y = 1 and the other with equation x + y = −1.
The correct answer is E;both statements together are still not sufficient.
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