Set 11: Systems of Equations

Choice C is correct. Choice C is the correct answer. Notice that the second equation is exactly double the first equation. Step 1: Multiply the first equation by 2: $$2(2 x + 3 y) = 2(6)$4x + 6 y = 12$$$ This is identical to the second equation! The two equations represent the same line. Step 2: Since both equations describe the same line, every point on that line is a solution. Strategic Tip: When one equation is a multiple of the other, the system has infinitely many solutions (coincident lines). Choice A is incorrect because the lines are identical, not parallel. Parallel lines with different y-intercepts have no solution. Choice B is incorrect because systems have exactly one solution only when the lines intersect at a single point (different slopes). Choice D is incorrect because two linear equations can never have exactly two solutions—they either don't intersect (0 solutions), touch at one point (1 solution), or are the same line (infinite solutions).