For infinitely many solutions of ((v+2)x-3y=6) and (10x-5y=10), what is (v)?

Correct answer: \(v=4\)

For two linear equations to have infinitely many solutions, \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\) must hold. Here, \(\frac{-3}{-5}=\frac{6}{10}=\frac{3}{5}\). Therefore, \(\frac{v+2}{10}=\frac{3}{5}\), giving \(v+2=6\) and hence \(v=4\). Option \(v=3\) is a close distractor, but it makes the first ratio \(\frac{1}{2}\), not \(\frac{3}{5}\). Exam tip: for infinitely many solutions, compare all three coefficient ratios.