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

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

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