For infinitely many solutions of ((p-2)x+7y=14) and (9x+21y=42), what is (p)?

Correct answer: (p=5)

The direct answer is option C: \(p=5\). Infinitely many solutions mean both equations describe exactly the same line. For equations \(a_1x+b_1y=c_1\) and \(a_2x+b_2y=c_2\), we need \(a_1/a_2=b_1/b_2=c_1/c_2\). Compare the second equation with the first: \(7/21=1/3\) and \(14/42=1/3\). Thus the first coefficient must satisfy \((p-2)/9=1/3\). Multiplying by 9 gives \(p-2=3\), so \(p=5\). Option A, \(p=3\), gives \((p-2)/9=1/9\), so the lines are not the same. Option B, \(p=4\), gives \(2/9\), also incorrect. Option C gives the required ratio and makes the first equation one-third of the second. Option D, \(p=7\), gives \(5/9\), so it fails. Memory cue: for infinitely many solutions, all three ratios must match, not just the two coefficient ratios.