For a fixed finite algebra 𝐀, we consider the decision problem SysTerm(𝐀): does a given system of term equations have a solution in 𝐀? This is equivalent to a constraint satisfaction problem (CSP) for a relational structure whose relations are the graphs of the basic operations of 𝐀. From the complexity dichotomy for CSP over fixed finite templates due to Bulatov [Bulatov, 2017] and Zhuk [Zhuk, 2017], it follows that SysTerm(𝐀) for a finite algebra 𝐀 is in P if 𝐀 has a not necessarily idempotent Taylor polymorphism and is NP-complete otherwise. More explicitly, we show that for a finite algebra 𝐀 in a congruence modular variety (e.g. for a quasigroup), SysTerm(𝐀) is in P if the core of 𝐀 is abelian and is NP-complete otherwise. Given 𝐀 by the graphs of its basic operations, we show that this condition for tractability can be decided in quasi-polynomial time.