The Dirac equation is considered in five-dimensional spaces with signatures (2,3), (4,1) and (0,5). The algebraic spinor formalism with the application of fermionic variables is used as the basis of real Clifford algebras and the module over this algebra. It is shown that solutions to the five-dimensional Dirac equation in spaces with signatures (2,3) and (4,1) can be expanded over solutions with zero value of the fifth component of the generalized momentum, and the equation is equivalent to an equation in four-dimensional spacetime.