A number is called y-smooth if all of its prime factors are bounded above by y. The set of y-smooth numbers below x forms a sparse subset of the integers below x as soon as x is sufficiently large in terms of y. If f_1, …, f_r \in Z[x_1,…,x_s] is a system of pairwise Q-linearly independent linear forms, one may ask how often these forms simultaneously take values in the set of y-smooth numbers when the variables x_i are all restricted to be of size about x. I will discuss asymptotic results on this counting problem. This talk is based on joint work with Mengdi Wang.