What is the value of (p) for (2x+py=10) and (4x+6y=20) to have infinitely many solutions?

Correct answer: \(p=3\)

For infinitely many solutions, both linear equations must represent the same line. Therefore, the ratios of corresponding coefficients must be equal: \(\frac{2}{4}=\frac{p}{6}=\frac{10}{20}\). Since \(\frac{2}{4}=\frac{1}{2}\), we get \(\frac{p}{6}=\frac{1}{2}\), so \(p=3\). If \(p=6\), the coefficients are not in the same ratio, so the equations do not represent the same line. Exam tip: for infinitely many solutions, check \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\).