Find the value of (a) for infinitely many solutions of (8x+ay=12) and (20x+10y=30).

Correct answer: \(a=4\)

For infinitely many solutions, the two linear equations must represent the same line; hence \(\frac{8}{20}=\frac{a}{10}=\frac{12}{30}\). Here, \(\frac{8}{20}=\frac{12}{30}=\frac{2}{5}\). Therefore, \(\frac{a}{10}=\frac{2}{5}\), which gives \(a=4\). For \(a=3,5\), or \(6\), the ratios of the corresponding coefficients are not equal. Exam tip: for infinitely many solutions, check that \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\).