How do you know when a system of equations is inconsistent?

1 Answer
Jim H
Mar 28, 2015

When you try to solve the system, you get an impossibility.
You get something like #3=8# or #x+5=x-2# (which would lead to #5=-2#

If you're working in the real numbers with nonlinear systems, you might instead get an imaginary solution.

(For example: #y=x^2+5# and #y=x+1#. By substitution: #x^2-x+4=0# but #b^2-4ac=(-1)^2-4(1)(4))# is negative.)

A system is inconsistent if, being a solution to one equation is inconsistent with being a solution of another equation in the system.

Being "inconsistent with" mean they can't both happen.
For example: being negative is inconsistent with being positive.
Being less than 4 is inconsistent with being greater than 9.

Being a solution to #y=3x+1# is inconsistent with being a solution to #y=3x-6#.
(#y# being one more than #3x# is inconsistent with #y# being 6 less than #3x#

The system:
#y=3x+1#
#y=3x-6#.
is inconsistent.

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