Question

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

Answer 1

Step 1. Theory of inconsistent system of equations:

At the point when you attempt to tackle the framework, you get a difficulty.

When getting something like $3=8$ or $x+5=x-2$ (which leads to $5=-2$)

Assuming that you're working in the genuine numbers with nonlinear frameworks, you could rather get a fanciful arrangement.

Example:

$y=x^2+5$ and $y=x+1$ By substitution $x^2-x+4=0$. But $b^2-4ac={\left(-1\right)}^2-4\left(1\right)\left(4\right)$ is negative.

A framework is inconsistent if, being an answer for one condition is conflicting with being an answer to one more condition in the framework.

Step 2. Take an example of being inconsistent.

Being "conflicting with" mean the two of them can't occur.

For example:

Being negative is conflicting with being positive and 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 more than $3x$ is inconsistent with $y$ being $6$ less than $3x$)

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

Hence, being negative is inconsistent with being positive.