Which one of the following pair of linear equations is/are inconsiste…

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

A pair \(a_1x+b_1y=c_1\), \(a_2x+b_2y=c_2\) is inconsistent (no solution) when \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\): the lines are parallel and never meet. If all three ratios are equal, the two equations give the same line and there are infinitely many solutions.

A. \(x-y=8,\ 6x-6y=32\) Inconsistent. The lines are parallel.

• Ratios of the x and y coefficients: \(\frac{1}{6}\) and \(\frac{-1}{-6}=\frac{1}{6}\), equal

• Ratio of the constants: \(\frac{8}{32}=\frac{1}{4}\), different

B. \(x+y=5,\ 4x+4y=20\) Consistent. The second equation is the first multiplied by 4, so it is the same line, with infinitely many solutions.

• All three ratios are equal: \(\frac{1}{4}\), \(\frac{1}{4}\) and \(\frac{5}{20}=\frac{1}{4}\)

C. \(4x-6y=16,\ 8x-12y=18\) Inconsistent. The lines are parallel.

• Ratios of the x and y coefficients: \(\frac{4}{8}=\frac{1}{2}\) and \(\frac{-6}{-12}=\frac{1}{2}\), equal

• Ratio of the constants: \(\frac{16}{18}=\frac{8}{9}\), different

D. \(4x-4y-4=0,\ 8x-8y-10=0\) Inconsistent. The lines are parallel.

• Ratios of the x and y coefficients: \(\frac{4}{8}=\frac{1}{2}\) and \(\frac{-4}{-8}=\frac{1}{2}\), equal

• Ratio of the constants: \(\frac{-4}{-10}=\frac{2}{5}\), different