Inconsistent equations

In mathematics and in particular in algebra, a linear or nonlinear system of equations is consistent if there is at least one set of values for the unknowns that satisfies every equation in the system—that is, that when substituted into each of the equations makes the equation hold true as an identity. In contrast, an equation system is inconsistent if there is no set of values for the unknowns that satisfies all of the equations.

If a system of equations is inconsistent, then it is possible to manipulate and combine the equations in such a way as to obtain contradictory information, such as 2 = 1, or x³ + y³ = 5 and x³ + y³ = 6 (which implies 5 = 6).

Both types of equation system, consistent and inconsistent, can be any of overdetermined (having more equations than unknowns), underdetermined (having fewer equations than unknowns), or exactly determined.

The system

has an infinite number of solutions, all of them having z = 1 (as can be seen by subtracting the first equation from the second), and all of them therefore having x+y = 2 for any values of x and y.

The nonlinear system

has an infinitude of solutions, all involving ${\displaystyle z=\pm {\sqrt {5}}.}$

Since each of these systems has more than one solution, it is an indeterminate system.

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