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

The correct answer is Option (3) → A, B and D only

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

A. \(x-y=6\), \(x-y=13\) Correct.

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

• Ratio of the constants: \(\frac{6}{13}\), different, so the lines are parallel (the same \(x-y\) cannot be both 6 and 13)

B. \(2x-10y=15\), \(7x-35y=25\) Correct.

• Ratios of the \(x\) and \(y\) coefficients: \(\frac{2}{7}\) and \(\frac{-10}{-35}=\frac{2}{7}\), equal

• Ratio of the constants: \(\frac{15}{25}=\frac{3}{5}\), different, so the lines are parallel

C. \(x-y=15\), \(5x-5y=75\) Incorrect.

• All three ratios are \(\frac{1}{5}\): the second equation is the first multiplied by 5

• The lines coincide, so the pair is consistent

D. \(7x-9y-3=0\), \(14x-18y-8=0\) Correct.

• Ratios of the \(x\) and \(y\) coefficients: \(\frac{7}{14}\) and \(\frac{-9}{-18}\), both \(\frac{1}{2}\)

• Ratio of the constants: \(\frac{-3}{-8}=\frac{3}{8}\), different, so the lines are parallel