Which one of the following pairs of linear equations is/are inconsist…

The correct answer is Option (2) → (B), (C) and (D) are inconsistent

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 are the same line and the pair is consistent, with infinitely many solutions.

(A) \(7x+y=5,\ 14x+2y=10\) Consistent (Incorrect). All three ratios are equal, so the second equation is the first one doubled: the same line.

• All three ratios: \(\frac{a_1}{a_2}=\frac{7}{14}=\frac{1}{2}\), \(\frac{b_1}{b_2}=\frac{1}{2}\), \(\frac{c_1}{c_2}=\frac{5}{10}=\frac{1}{2}\)

(B) \(2x-3y=7,\ 4x-6y=12\) Inconsistent (Correct). The \(x\) and \(y\) ratios agree but the constants do not, so the lines are parallel.

• Ratios of the \(x\) and \(y\) coefficients: \(\frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2}\), \(\frac{b_1}{b_2}=\frac{-3}{-6}=\frac{1}{2}\)

• Ratio of the constants: \(\frac{c_1}{c_2}=\frac{7}{12}\), which is not \(\frac{1}{2}\)

(C) \(x-y=10,\ 3x-3y=16\) Inconsistent (Correct). Again only the constants break the pattern, so the lines are parallel.

• Ratios of the \(x\) and \(y\) coefficients: \(\frac{a_1}{a_2}=\frac{1}{3}\), \(\frac{b_1}{b_2}=\frac{-1}{-3}=\frac{1}{3}\)

• Ratio of the constants: \(\frac{c_1}{c_2}=\frac{10}{16}\), which is not \(\frac{1}{3}\)

(D) \(2x-2y-2=0,\ 4x-4y-5=0\) Inconsistent (Correct). In the form \(ax+by+c=0\) the same test applies, and the lines are parallel.

• Ratios of the \(x\) and \(y\) coefficients: \(\frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2}\), \(\frac{b_1}{b_2}=\frac{-2}{-4}=\frac{1}{2}\)

• Ratio of the constants: \(\frac{c_1}{c_2}=\frac{-2}{-5}=\frac{2}{5}\), which is not \(\frac{1}{2}\)