In contrast, a linear or non linear equation system is called inconsistent if there is no set of values for the unknowns that satisfies all of the equations.
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 system has no solutions, as can be seen by subtracting the first equation from the second to obtain the impossible 0 = 1.
The non-linear system has no solutions, because if one equation is subtracted from the other we obtain the impossible 0 = 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 of z can be chosen and values of x and y can be found to satisfy the first two (and hence the third) equations.
Likewise, is an inconsistent system because the first equation plus twice the second minus the third contains the contradiction 0 = 2.
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.
As can be seen from the above examples, consistency versus inconsistency is a different issue from comparing the numbers of equations and unknowns.