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In the xy-plane, line l and line k intersect at the point (16/5, 12/5). What is the slope of line l? (1) The product of the slopes of line l and line k is –1. (2) Line k passes through the origin.

Line \(l\) passes through the point (16/5, 12/5). If we knew some other point through which line \(l\) passes then we would be able to calculate the slope: the slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line.

(1) The product of the slopes of line \(l\) and line \(k\) is -1 --> line \(l\) and \(k\) are perpendicular to each other (the two lines are perpendicular if and only the product of their slopes is -1). Not, sufficient, as we can have infinite # of perpendicular lines passing through some point (16/5, 12/5).

(2) Line \(k\) passes through the origin --> we have the second point for line \(k\), so we can calculate the slope of \(k\), but we don't know the relationship between the lines \(l\) and \(k\). Not sufficient.

(1)+(2) We can calculate the slope of \(k\) and we know that the product of the slopes of \(l\) and \(k\) is -1, so we can calculate the slope of line \(l\) too. Sufficient.

Answer: C.

For more on these issues check Coordinate Geometry chapter of Math Book (link in my signature).

(1) Insufficient. First, you must know that perpendicular lines have negative reciprocal slopes. We know that the product of the slopes of l and k is -1, therefore they are perpendicular.

(2) Insufficient. We can now solve for the slope of k, but we know nothing about l.

(1+2) Sufficient, Since we know the slope of k, then we know the slope of l is -1/k.

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