Which statement is correct for (6x+7y=23) and (12x+13y=45)?

Correct answer: \(\frac{6}{12}\ne\frac{7}{13}\), so there is a unique solution

Here, \(a_1=6, a_2=12, b_1=7, b_2=13\). Thus, \(\frac{a_1}{a_2}=\frac{6}{12}=\frac{1}{2}\), whereas \(\frac{b_1}{b_2}=\frac{7}{13}\); the two ratios are unequal. For a pair of linear equations, the condition \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\) indicates a unique solution. In fact, solving the equations gives \(y=1\) and \(x=8/3\), so the value \(y=2\) in option A is incorrect. In an exam, compare the ratios of the coefficients of x and y first; if they are unequal, the pair has a unique solution.