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

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

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Answer 1 · 2015-03-28T17:02:44

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

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

(For example: $y = x^{2} + 5$ $y=x^2+5$ and $y = x + 1$ $y=x+1$ . By substitution: $x^{2} - x + 4 = 0$ $x^2-x+4=0$ but $b^{2} - 4 a c = {(- 1)}^{2} - 4 (1) (4))$ $b^2-4ac=(-1)^2-4(1)(4))$ 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 $y = 3 x + 1$ $y=3x+1$ is inconsistent with being a solution to $y = 3 x - 6$ $y=3x-6$ .
( $y$ $y$ being one more than $3 x$ $3x$ is inconsistent with $y$ $y$ being 6 less than $3 x$ $3x$

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

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

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