Question

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

Answer 1

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.