Problem 4 · Concurrency with a distinctness check · 4 points
Made by AI and not reviewed by a real person.
For which value of k do y = kx + 1, y = 2x + k − 1, and y = x + 3 form three distinct lines through one point?
- 1
- -1
- 4
- 2
- 3
Show a hint
- Two of the lines always have different slopes. Find where they meet, then check whether the third line passes through that point.
- The lines with slopes 2 and 1 meet at x = 4 − k. After finding possible values of k, check that the lines really are distinct.
Show the answer and solution
Answer: E — 3
Setting 2x + k − 1 = x + 3 gives x = 4 − k. For the first line to pass through that point, k(4 − k) + 1 = (4 − k) + 3, which simplifies to (k − 2)(k − 3) = 0. The tempting value 2 makes the first two lines identical, not distinct. At k = 3, the three lines have different slopes and all pass through (1, 4), so the answer is 3.