For infinitely many solutions of (jx+5y=10) and (14x+7y=14), what is (j)?

Correct answer: \(j=10\)

For infinitely many solutions, the two linear equations must represent the same line; hence \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\). Here, \(\frac{5}{7}=\frac{10}{14}\), so \(\frac{j}{14}=\frac{5}{7}\). Therefore, \(j=10\). For example, if \(j=8\), then \(\frac{j}{14}\neq\frac{5}{7}\), so the equations cannot have infinitely many solutions. Exam tip: for infinitely many solutions, verify that all three ratios are equal.