We present a simple semi-streaming algorithm for $(1-ε)$-approximation of bipartite matching in $O(\log{\!(n)}/ε)$ passes. This matches the performance of state-of-the-art "$ε$-efficient" algorithms -- the ones with much better dependence on $ε$ albeit with some mild dependence on $n$ -- while being considerably simpler. The algorithm relies on a direct application of the multiplicative weight update method with a self-contained primal-dual analysis that can be of independent interest. To show case this, we use the same ideas, alongside standard tools from matching theory, to present an equally simple semi-streaming algorithm for $(1-ε)$-approximation of weighted matchings in general (not necessarily bipartite) graphs, again in $O(\log{\!(n)}/ε)$ passes.