Question

What defines an inconsistent linear system? Can you solve an inconsistent linear system?

Answer 1

Inconsistent system of equations is, by definition, a system of equations for which there is no set of unknown values that transforms it into a set of identities.
It is unsolvable by definiton.

Explanation:

Example of an inconsistent single linear equation with one unknown variable:
`2x+1 = 2(x+2)`
Obviously, it is fully equivalent to
`2x+1 = 2x+4`
or
`1=4`,
which is not an identity, there is no such `x` that transforms the initial equation into an identity.

Example of an inconsistent system of two equations:
`x+2y=3`
`3x-1=4-6y`
This system is equivalent to
`x+2y=3`
`3x+6y=5`
Multiply the first equation by `3`. The result is
`3x+6y=9`
It is, obviously, inconsistent with the second equation, where the same expression that contains `x` and `y` on the left has a different value (`5`) on the right.
Hence, the system has no solutions.

So, we can say that an inconsistent system has no solutions. This follows from the definition of inconsistency.