What is the correct ratio relation for the equations (10x-7y+31=0) and (20x-14y+65=0)?

Correct answer: \(\frac{10}{20}=\frac{-7}{-14}\ne\frac{31}{65}\)

Here, \(a_1=10, b_1=-7, c_1=31\) and \(a_2=20, b_2=-14, c_2=65\). We get \(\frac{a_1}{a_2}=\frac{10}{20}=\frac{1}{2}\) and \(\frac{b_1}{b_2}=\frac{-7}{-14}=\frac{1}{2}\), but \(\frac{c_1}{c_2}=\frac{31}{65}\ne\frac{1}{2}\). Thus, \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\), so the pair is inconsistent and has no solution because the corresponding lines are parallel. Exam tip: Compare the ratios of the coefficients of both variables first, and then compare them with the ratio of the constants.