  We study the thresholds for the property of containing a solution to a linear
homogeneous system in random sets. We expand a previous sparse Sz\'emeredi-type
result of Schacht to the broadest class of matrices possible. We also provide a
shorter proof of a sparse Rado result of Friedgut, R\"odl, Ruci\'nski and
Schacht based on a hypergraph container approach due to Nenadov and Steger.
Lastly we further extend these results to include some solutions with repeated
entries using a notion of non-trivial solutions due to R\'uzsa as well as Ru\'e
et al.
