What is the correct condition on (t) for the equations (5x+9y=64) and (15x+27y=t) to be inconsistent?

Correct answer: (t\ne192)

The direct answer is option B: \(t\ne192\). For two equations \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\), inconsistency occurs when the coefficient ratios are equal but the constant ratio is different. Here the second left side is exactly three times the first: \(15=3\times5\) and \(27=3\times9\). Therefore, to represent the same line, its right side would also have to be \(3\times64=192\). If \(t=192\), the equations are the same line and have infinitely many solutions. If \(t\ne192\), they have the same left side direction but different positions, so they are parallel and never meet. Option A is wrong because it gives the dependent case, not inconsistency. Option B is correct because every value other than 192 makes the constants incompatible. Option C is wrong because 64 is not the required threefold value. Option D is wrong because 128 is also not 192. Memory cue: when one complete equation is multiplied by 3, multiply its constant by 3 too; any other constant gives parallel lines.