Which one of the following pairs of linear equations is/are inconsistent?
(A) \(7x+2y=3\), \(14x+4y=5\)
(B) \(2x−3y=7\), \(4x−6y=12\)
(C) \(x−y=10\), \(3x−3y=16\)
(D) \(2x−2y=2\), \(4x−4y=5\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B), (C) and (D)
A pair \(a_1x+b_1y=c_1\), \(a_2x+b_2y=c_2\) is inconsistent (has no solution) when the two lines are parallel and distinct, that is, when \(\displaystyle\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\). Each pair is tested with these ratios.
(A) \(7x+2y=3\), \(14x+4y=5\) Correct.
\(x\) and \(y\) ratios: \(\frac{7}{14}\) and \(\frac{2}{4}\), both \(\frac{1}{2}\)
Constant ratio \(\frac{3}{5}\) differs, so the lines are parallel and distinct
(B) \(2x-3y=7\), \(4x-6y=12\) Correct.
\(x\) and \(y\) ratios: \(\frac{2}{4}\) and \(\frac{-3}{-6}\), both \(\frac{1}{2}\)
Constant ratio \(\frac{7}{12}\) differs, so the lines are parallel and distinct
(C) \(x-y=10\), \(3x-3y=16\) Correct.
\(x\) and \(y\) ratios: \(\frac{1}{3}\) and \(\frac{-1}{-3}\), both \(\frac{1}{3}\)
Constant ratio \(\frac{10}{16}\) differs, so the lines are parallel and distinct
(D) \(2x-2y=2\), \(4x-4y=5\) Correct.
\(x\) and \(y\) ratios: \(\frac{2}{4}\) and \(\frac{-2}{-4}\), both \(\frac{1}{2}\)
Constant ratio \(\frac{2}{5}\) differs, so the lines are parallel and distinct
Tip: Compare the three ratios once: all three equal means the same line (infinitely many solutions), only the first two equal means no solution, and the first two unequal means a single solution.