What is the value of (p) for (px-6y=18) and (2x-3y=9) to have infinitely many solutions?

Correct answer: \(p=4\)

For infinitely many solutions, both equations must represent the same line. Therefore, the ratios of corresponding coefficients must be equal: \(\frac{p}{2}=\frac{-6}{-3}=\frac{18}{9}=2\). Hence \(\frac{p}{2}=2\), so \(p=4\). If \(p=2\), the ratio of the coefficients of \(x\) would not match the other ratios. Exam tip: for infinitely many solutions, check \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\).