What is the value of k?

Question

A museum exhibit on textiles displays a loom pattern on a coordinate grid. One warp thread is represented by the graph of 6x + ky = 30, where k is a constant, and one weft thread is represented by the graph of 4x - 3y = 9. In the pattern, the two threads are perpendicular.

Accepted Answer

D. 8. The weft thread 4x - 3y = 9 gives y = (4/3)x - 3, slope 4/3. The warp thread 6x + ky = 30 gives y = -(6/k)x + 30/k, slope -6/k. Perpendicular lines require -6/k = -3/4, the negative reciprocal of 4/3, so -24 = -3k and k = 8. Choice A comes from setting -6/k equal to 4/3, making the lines parallel rather than perpendicular. Choice B comes from using -4/3 (sign changed but no reciprocal taken). Choice C comes from a sign error when solving -6/k = -3/4, which would give the slope 3/4 instead of -3/4.

Answer

A. -9/2 — Setting -6/k = 4/3 makes the two threads parallel. For perpendicular threads the warp slope must be the negative reciprocal, -3/4, which gives k = 8.

Answer

B. 9/2 — You negated 4/3 but didn't take the reciprocal. The negative reciprocal of 4/3 is -3/4, and solving -6/k = -3/4 gives k = 8.

Answer

C. -8 — You had the right slope relationship but slipped on a sign while solving: -6/k = -3/4 gives -24 = -3k, so k = 8, not -8. Using slope 3/4 instead of -3/4 is what produces -8.

Algebra practice question

Answer: D. 8

Why the other answers are wrong

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