It has recently been shown that several computational models, such as trellis automata, recursive functions and Turing machines, admit characterization by resolved systems of language equations with different sets of language-theoretic operations. This paper investigates how simple the systems of equations from the computationally universal types could be while still retaining their universality. It is proved that the universality and the associated hardness of decision problems begins at one-variable equations. Additionally, it is shown that language equations with added quotient with regular languages can specify every set representable in first-order Peano arithmetic.