What is \(r\) for infinitely many solutions of \(rx+\frac{3}{2}y=6\) and \(8x+6y=24\)?

Correct answer: \(r=2\)

For infinitely many solutions, the two linear equations must represent the same line; hence \(\frac{r}{8}=\frac{\frac{3}{2}}{6}=\frac{6}{24}\). The last two ratios are \(\frac{1}{4}\), so \(\frac{r}{8}=\frac{1}{4}\) gives \(r=2\). If \(r=4\), then \(\frac{r}{8}=\frac{1}{2}\), which is not equal to the other ratios. Exam tip: for infinitely many solutions, verify \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\).