Answer: \((x, y) = (3, 2)\)
\(99x + 101y = 499\) ... (I) and \(101x + 99y = 501\) ... (II). The coefficients are interchanged, so add and subtract the equations.
Adding (I) and (II):\[200x + 200y = 1000\]\[x + y = 5 \quad \text{(III)}\]
Subtracting (I) from (II):\[2x - 2y = 2\]\[x - y = 1 \quad \text{(IV)}\]
Adding (III) and (IV): \(2x = 6\), so \(x = 3\). Then from (III), \(y = 5 - 3 = 2\).
Check in (I): \(99(3) + 101(2) = 297 + 202 = 499\), which is true.