Inmathematics and particularly inalgebra, asystem of equations (eitherlinear ornonlinear) is calledconsistent if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, whensubstituted into each of the equations, they make each equation hold true as anidentity. In contrast, a linear or non linear equation system is calledinconsistent if there is no set of values for the unknowns that satisfies all of the equations.[1][2]
If a system of equations is inconsistent, then the equations cannot be true together leading to contradictory information, such as the false statements2 = 1, or and (which implies5 = 6).
Both types of equation system, inconsistent and consistent, can be any ofoverdetermined (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 havingz = 1 (as can be seen by subtracting the first equation from the second), and all of them therefore havingx +y = 2 for any values ofx andy.
The nonlinear system
has an infinitude of solutions, all involving
Since each of these systems has more than one solution, it is anindeterminate system .
The system
has no solutions, as can be seen by subtracting the first equation from the second to obtain the impossible0 = 1.
The non-linear system
has no solutions, because if one equation is subtracted from the other we obtain the impossible0 = 3.
The system
has exactly one solution:x = 1,y = 2
The nonlinear system
has the two solutions(x, y) = (1, 0) and(x, y) = (0, 1), while
has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value ofz can be chosen and values ofx andy can be found to satisfy the first two (and hence the third) equations.
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 impossible0 = 2.
Likewise,
is an inconsistent system because the first equation plus twice the second minus the third contains the contradiction0 = 2.
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 give the same information as each other (as can be seen by multiplying through the first equation by either 3 or 7). Any value ofy is part of a solution, with the corresponding value ofx being7 – 2y.
The nonlinear system
has the three solutions(x, y) = (1, –1), (–1, 1), (1, 1).
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.
The system
is inconsistent because the sum of the first two equations contradicts the third one.
As can be seen from the above examples, consistency versus inconsistency is a different issue from comparing the numbers of equations and unknowns.
A linear system is consistentif and only if itscoefficient matrix has the samerank as does itsaugmented matrix (the coefficient matrix with an extra column added, that column being thecolumn vector of constants).