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The number of values of k for which the system of linear equations, (k + 2)x + 10y = k, kx + (k + 3)y = k − 1 has no solution, is

1. Infinitely many
2. 3
3. 1
4. 2

Correct answer: 1

Solution

For no solution, determinant (k+2)(k+3)-10k = k^2-5k+6 = 0 gives k=2 or k=3. At k=2 all coefficient ratios equal (consistent -> infinitely many solutions). At k=3 the coefficient ratios match (5/3) but differ from the constant ratio (3/2), giving no solution. Hence exactly 1 value of k.

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