Consistent and inconsistent equations

In mathematics and particularly in algebra, a system of equations (either linear or nonlinear) is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, when substituted into each of the equations, they make each equation hold true as an identity. 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. 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 and (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 Since each of these systems has more than one solution, it is an indeterminate system. 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. 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.