Which of the following pair of linear equations is inconsistent?

(A) $x - y = 5; 3x-3y = 10$
(B) $2x + 3y = 4; 4x+6y=8$
(C) $9x+6y= 6; 3x + 2y = 3$
(D) $2x+5y= 2; 6x-15y = 4$

Choose the correct answer from the options given below:

(A) and (C) only

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

To check inconsistency of a pair of linear equations:

For
$a_1x + b_1y = c_1$ ​
$a_2x + b_2y = c_2$​

The pair is inconsistent if

$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

(A) $x - y = 5;\; 3x - 3y = 10$

$\frac{1}{3} = \frac{-1}{-3} \neq \frac{5}{10}$​

Inconsistent

(B) $2x + 3y = 4;\; 4x + 6y = 8$

$\frac{2}{4} = \frac{3}{6} = \frac{4}{8}$​

Consistent (infinitely many solutions)

(C) $9x + 6y = 6;\; 3x + 2y = 3$

$\frac{9}{3} = \frac{6}{2} \neq \frac{6}{3}$

Inconsistent

(D) $2x + 5y = 2;\; 6x - 15y = 4$

$\frac{2}{6} \neq \frac{5}{-15}$​

Consistent (unique solution)

Correct Answer: (A) and (C) only