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At the result, if you get $0=-6$, does that mean that there is no solution. If you get $5=5$ or $6=6$, does that mean any value of $x$ (besides those values that are restricted) are solutions?

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  • $\begingroup$ It depends entirely on context. Can you give an example of such questions? Certainly if having assumed that there exists a solution and following valid steps you find that it implies that $0=-6$, then that means that your assumption was incorrect and that no solutions exist. For example, "find all real numbers $x$ that satisfy the equation $x=x-6$" $\endgroup$ – JMoravitz Dec 5 '15 at 21:41
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Yes. When solving linear equations and you obtain a false result, such as $4=5$ or $0=-6$, that means there are no solutions. If the result is always true, i.e. $5=5$ then your linear equations have infinitely many solutions.

For example:

$y = x + 5$ and $x = y - 5$ which are the same equations and thus have infinitely many solutions.

Adding these two together:

$y + x = x + y + 5 - 5$

Of course, everything cancels so that $0=0$, but you can also leave the $5$'s and that shows you $5 = 5$

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