What is a consistent linear system?

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Answer 1 · 2016-05-03T08:44:54

A consistent linear system is a system of linear equations with at least one set of values satisfying all equations.

A system of linear equations is said to be consistent if there is a solution which satisfies all of the equations. For example,

#{(x+y = 1), (x+2y = 5):}#

has the solution

#{(x = -3), (y = 4):}#

and thus is consistent.

The system

#{(x+y = 1), (2x+2y = 2):}#

has infinitely many solutions, as any #(x,y)# pair will work so long as #y = -x+1#. As such, it is also a consistent system.

However, the following system is not consistent

#{(x+y = 1), (x+y = 2):}#

as there is clearly no pair of values #(x,y)# which fulfill both equations.