No-Solution Condition for a Linear System

A 2×2 system has no unique solution when its determinant is zero: (1)(k) − (2)(3) = 0, so k = 6. With k = 6 the lines are parallel: tripling the first equation gives 3x + 6y = 12, but the second says 3x + 6y = 5. The same left side cannot equal both 12 and 5, so there is no solution rather than infinitely many. TI-36X Pro tip: enter each answer choice for k in 2nd, sys-solv, 2x2 LIN EQUs. Only k = 6 shows No Solution Found, which confirms the lines are parallel.