For what value of k do the three lines 3x + 4y = 5, 5x + 4y = 4, and kx + 4y = 6 meet at a single point?

Correct answer: 1

Solution

Subtracting 3x+4y=5 from 5x+4y=4 gives 2x = -1, so x = -1/2. Substituting: 4y = 5 - 3(-1/2) = 5 + 3/2 = 13/2, so y = 13/8. Substituting into kx+4y=6: k(-1/2) + 4(13/8) = 6 => -k/2 + 13/2 = 6 => -k/2 = 6 - 13/2 = -1/2 => k = 1.

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