What is the value of (t) for infinitely many solutions of (tx+9y=6) and (20x+15y=10)?

Correct answer: \(t=12\)

For infinitely many solutions, the two linear equations must represent the same line. Hence, the ratios of corresponding coefficients and constants must be equal: \(\frac{t}{20}=\frac{9}{15}=\frac{6}{10}\). Since \(\frac{9}{15}=\frac{6}{10}=\frac{3}{5}\), we get \(\frac{t}{20}=\frac{3}{5}\). Therefore, \(t=12\). For \(t=10\), \(\frac{t}{20}=\frac{1}{2}\), which is not equal to \(\frac{3}{5}\). Exam tip: use \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\) to check for infinitely many solutions.