Set 12: Systems of Equations (Intermediate)

Choice D is correct. Choice D is the correct answer. For no solution, the system must represent parallel lines (same coefficients on $x$ and $y$, but different constants). Step 1: Notice that the second equation has coefficients that are exactly double the first: - First equation: $3x - 2 y = 7$ - Second equation: $6x - 4 y = k$ Step 2: If we multiply the first equation by 2: $$2(3 x - 2 y) = 2(7)6x - 4 y = 14$$$ Step 3: For the lines to be parallel but not identical: - If k = 14$: the lines are identical → infinitely many solutions - If$k \neq 14$: the lines are parallel → no solution Strategic Tip: Parallel lines occur when left sides are proportional but right sides are not. Choice A is incorrect because$k = 7$gives no solution (not 14), but it's not the only value. Choice B is incorrect because$k = 10$gives no solution, but it's not the only value. Choice C is incorrect because$k = 14$ gives infinitely many solutions, not no solution.