System of linear equations: Difference between revisions

Revision as of 10:32, 14 December 2008

Revision as of 06:50, 20 December 2008

Line 127:

For example, the equations

:<math>3x+2y=6\;\;\;\;\text{and}\;\;\;\;3x+2y=12</math>

are inconsistent. In attempting to find a solution, we tacitly assume that there '''is''' a solution; that is, we assume that we can subtract the first equation from the second since, if there is a solution, the value of ''x'' in the first equation must be the same as the value of ''x'' in the second equation (the same is assumed to simultaneously be true for the value of ''y'' in both equations). SubtractingApplying the firstsubstitution equationproperty from(for the second3x+2y) yields the equation 06&nbsp;=&nbsp;612, which is a false statement. This therefore contradicts our assumption that the system had a solution and we conclude that our assumption was false; that is, the system in fact has '''no solution'''. The graphs of these equations on the ''xy''-plane are a pair of [[parallel (geometry)|parallel]] lines.

It is possible for three linear equations to be inconsistent, even though any two of the equations are consistent together. For example, the equations