What is the correct ratio relation for the equations (12x-5y+37=0) and (24x-10y+79=0)?

Correct answer: \(\frac{12}{24}=\frac{-5}{-10}\ne\frac{37}{79}\)

Here, \(a_1=12, b_1=-5, c_1=37\) and \(a_2=24, b_2=-10, c_2=79\). Thus, \(\frac{a_1}{a_2}=\frac{12}{24}=\frac{1}{2}\) and \(\frac{b_1}{b_2}=\frac{-5}{-10}=\frac{1}{2}\), whereas \(\frac{c_1}{c_2}=\frac{37}{79}\ne\frac{1}{2}\). Therefore, \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\), so the two lines are parallel and distinct, giving no solution. Exam tip: When the first two coefficient ratios are equal but the constant-term ratio is different, the pair has no solution.