Which of the following pair of linear equations is inconsistent?(A) x…

The correct answer is Option (2) → (A) and (C) only

A pair \(a_1x+b_1y=c_1\), \(a_2x+b_2y=c_2\) is inconsistent (has no solution) when the two lines are parallel, that is, when \(\displaystyle\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\). If all three ratios are equal, the lines coincide; if the first two differ, the lines cross at one point.

(A) \(x-y=5;\ 3x-3y=10\) Correct. \(\frac{1}{3}=\frac{-1}{-3}\), but \(\frac{5}{10}=\frac{1}{2}\) is different, so the lines are parallel and the pair is inconsistent.

(B) \(2x+3y=4;\ 4x+6y=8\) Incorrect. All three ratios are \(\frac{1}{2}\): the second equation is the first doubled, so the lines coincide and there are infinitely many solutions.

(C) \(9x+6y=6;\ 3x+2y=3\) Correct. \(\frac{9}{3}=\frac{6}{2}=3\), but \(\frac{6}{3}=2\), so the lines are parallel and the pair is inconsistent.

(D) \(2x+5y=2;\ 6x-15y=4\) Incorrect. \(\frac{2}{6}=\frac{1}{3}\) but \(\frac{5}{-15}=-\frac{1}{3}\); the first two ratios differ, so the lines cross at one point and the pair has a unique solution.