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.
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.
Simple examples
Underdetermined and consistent
The system
has an infinitude 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.
Underdetermined and inconsistent
The system
has no solutions, as can be seen by subtracting the first equation from the second to obtain the impossible 0 = 1.
Exactly determined and consistent
The system
has exactly one solution: x = 1, y= 2.
Exactly determined and inconsistent
The system
has no solutions; the inconsistency can be seen by multiplying the first equation by 4 and subtracting the second equation to obtain the impossible 0 = 2.
Overdetermined and consistent
The system
has a solution, x = –1, y = 4, because the first two equations do not contradict each other and the third equation is redundant (since it contains the same information as can be obtained from the first two equations by multiplying each through by 2 and summing them).
The system
has an infinitude of solutions since all three equations say the same thing as each other (as can be seen by multiplying through the first equation by either 3 or 7). Any value of y is part of a solution, with the corresponding value of x being 7–2y.
Overdetermined and inconsistent
The system
is inconsistent because the last equation contradicts the information embedded in the first two, as seen by multiplying each of the first two through by 2 and summing them.
Criteria for consistency
As can be seen from the above examples, consistency versus inconsistency is a different issue from being underdetermined or exactly determined versus being overdetermined.
Linear systems
A linear system is consistent if and only if its coefficient matrix has the same rank as does its augmented matrix (the coefficient matrix with an extra column added, that column being the column vector of constants).