Answer: \((m, n) = (5, 2)\)
Let \(5m - 3n = 19\) ... (I) and \(m - 6n = -7\) ... (II).
Multiply (I) by 2: \(10m - 6n = 38\)
Subtract the equations to eliminate \(n\):\[9m = 45\]\[m = 5\]
Substitute \(m = 5\) in (I): \(-3n = 19 - 25 = -6\), so \(n = 2\).
Check in (II): \(1(5) + (-6)(2) = -7\), which is true.