Which of the following pairs of linear equations is/are inconsistent?…

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

A pair of linear equations is inconsistent (has no solution) when the two lines are parallel, that is, when \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\), with each equation written as \(ax+by=c\).

A. \(x-y=8;\ 3x-3y=16\) Correct. The lines are parallel.

• Ratios of the coefficients of \(x\) and \(y\): \(\frac{1}{3}\) and \(\frac{-1}{-3}=\frac{1}{3}\), equal

• Ratio of the constant terms: \(\frac{8}{16}=\frac{1}{2}\), different

B. \(3x+2y=5;\ 2x-3y=7\) Incorrect. Here \(\frac{3}{2}\ne\frac{2}{-3}\), so the lines cross at one point and the pair has a unique solution.

C. \(2x-2y-2=0;\ 4x-4y-5=0\) Correct. Written as \(2x-2y=2\) and \(4x-4y=5\), the lines are parallel.

• Ratios of the coefficients of \(x\) and \(y\): \(\frac{2}{4}\) and \(\frac{-2}{-4}\), both \(\frac{1}{2}\)

• Ratio of the constant terms: \(\frac{2}{5}\), different

D. \(2x-3y=8;\ 4x-6y-5=9\) Correct. The second equation is \(4x-6y=14\), and the lines are parallel.

• Ratios of the coefficients of \(x\) and \(y\): \(\frac{2}{4}\) and \(\frac{-3}{-6}\), both \(\frac{1}{2}\)

• Ratio of the constant terms: \(\frac{8}{14}=\frac{4}{7}\), different