If 12x − 6y = 42 and 2x − y = t are inconsistent, what is the correct condition on t?

Correct answer: t ≠ 7

The equations have the form 12x − 6y = 42 and 2x − y = t. Their x-coefficient ratio is 12/2 = 6, and their y-coefficient ratio is (−6)/(−1) = 6. Thus the corresponding lines have equal slopes. To be inconsistent, they must be different parallel lines, so the constant ratio must not equal the same value: 42/t ≠ 6. Solving the equality 42/t = 6 gives 42 = 6t and therefore t = 7. This value would make the equations dependent, because multiplying 2x − y = 7 by 6 gives 12x − 6y = 42, producing infinitely many solutions. Every value t other than 7 makes the lines distinct and parallel, with no common solution. Hence option B, t ≠ 7, is correct.