Let x be the flat fee and 0.99y be the cost of downloading 27 songs. We know that:
$$x + 0.99y = 42.72$$
Substituting y = 27, we get:
$$x + 0.99 * 27 = 42.72$$
$$x + 26.73 = 42.72$$
Subtracting 26.73 from both sides, we get:
$$x = 42.72 - 26.73$$
$$x = 15.99$$
Therefore, the flat fee is $15.99.