Which of the following pair of linear equations is / are inconsistent…

The correct answer is Option (4) → (A) and (C) only

For \(a_1x+b_1y=c_1\) and \(a_2x+b_2y=c_2\), the pair is inconsistent (it 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.

(A) x − y = 5; 3x − 3y = 10 Correct. The lines are parallel, so the pair is inconsistent:

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

• Ratio of the constants: \(\frac{5}{10}=\frac{1}{2}\), different

(B) 2x + 3y = 4; 4x + 6y = 8 Incorrect. The second equation is the first one doubled, so both give the same line, with infinitely many solutions:

• All three ratios: \(\frac{2}{4}\), \(\frac{3}{6}\) and \(\frac{4}{8}\), each one half

(C) 9x + 6y = 6; 3x + 2y = 3 Correct. The lines are parallel, so the pair is inconsistent:

• Ratios of the \(x\) and \(y\) coefficients: \(\frac{9}{3}\) and \(\frac{6}{2}\), both 3

• Ratio of the constants: \(\frac{6}{3}\), which is 2, different

(D) 2x + 5y = 2; 6x − 15y = 4 Incorrect. The lines cross at one point, so there is a unique solution:

• Ratios of the \(x\) and \(y\) coefficients: \(\frac{2}{6}\), which is one third, and \(\frac{5}{-15}\), which is minus one third, different