Let x be the distance between the center of the Earth and the center of gravity of the Earth-Moon system. Taking moments around the center of gravity:
5.98 × 1024 (x) = 7.35 × 1022 (384.4 × 103 - x)
Solving for x:
5,98 × 1024 (x) = 2825.3 × 1025 - 7.35 × 1022 (x)
598 × 1022(x) + 7.35 × 1022(x) = 2825.3 × 1025
605.35 × 1022(x) = 2825.3 × 1025
x = (2825.3 x 1025)/(605.35 x 1022)
x = 4.667 x 103km