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

The correct answer is Option (4) → (B), (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 the pair is consistent, with infinitely many solutions.

(A) \(2x-2y=10\), \(3x-3y=15\) Consistent (Incorrect). All three ratios are equal, so both equations reduce to \(x-y=5\), the same line.

• Ratios of the \(x\) and \(y\) coefficients: \(\frac{2}{3}\) and \(\frac{2}{3}\)

• Ratio of the constants: \(\frac{10}{15}\), which is also \(\frac{2}{3}\)

(B) \(3x-y=7\), \(6x-2y=18\) Inconsistent (Correct). The lines are parallel.

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

• Ratio of the constants: \(\frac{7}{18}\), which is not one half

(C) \(x-y=15\), \(5x-5y=12\) Inconsistent (Correct). The lines are parallel.

• Ratios of the \(x\) and \(y\) coefficients: \(\frac{1}{5}\) and \(\frac{1}{5}\)

• Ratio of the constants: \(\frac{15}{12}\), which is not \(\frac{1}{5}\)

(D) \(7x-3y=3\), \(14x-6y=8\) Inconsistent (Correct). The lines are parallel.

• Ratios of the \(x\) and \(y\) coefficients: \(\frac{7}{14}\) and \(\frac{3}{6}\), both one half

• Ratio of the constants: \(\frac{3}{8}\), which is not one half