Indeterminate system

In mathematics, particularly in algebra, an indeterminate system is a system of simultaneous equations (e.g., linear equations) which has more than one solution (sometimes infinitely many solutions).^[1] In the case of a linear system, the system may be said to be underspecified, in which case the presence of more than one solution would imply an infinite number of solutions (since the system would be describable in terms of at least one free variable^[2]), but that property does not extend to nonlinear systems (e.g., the system with the equation ${\displaystyle x^{2}=1}$).

An indeterminate system by definition is consistent, in the sense of having at least one solution.^[3] For a system of linear equations, the number of equations in an indeterminate system could be the same as the number of unknowns, less than the number of unknowns (an underdetermined system), or greater than the number of unknowns (an overdetermined system). Conversely, any of those three cases may or may not be indeterminate.