# Properties of an inconsistent system of linear equations

Given the inconsistent system of linear equations

$$y = 3x-1$$

$$y = 3x+1$$

What sort of properties could be expressed about the system, other than the obvious like "the lines are parallel, slope is equal," etc.?

The direction I'm going here is that it seems there is some way to express this as one function like $f(b)=3x+b$ where $b= \pm 1$, but that obviously can't graph correctly, since there is no $y$ involved. Any ideas?

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That's not inconsistent: $x=0,\;y=1$ is a perfectly good solution (and unique, too). –  Henning Makholm Oct 28 '11 at 0:38
Fixed. It was supposed to differ only by y-intercept –  wtfsven Oct 28 '11 at 0:45
You could be interested in [this][1] if it is not too complicated... [1]: math.stackexchange.com/questions/818997/… –  Surb Jun 5 '14 at 9:16

The question isn't clear to me, so maybe I'm barking up the wrong tree, but the graph of the single equation $$(y-3x)^2=1$$ is precisely those two lines.
We could say that the matrix that represents the coefficients has determinant $0$.
Notice that if you're trying to define "one" function, then you can't have a ' $\pm$ ', at least not in the way you were trying to use it. Part of the definition of a function requires that it take exactly one value for each point in the domain.
But the matrix of the system can have determinant $0$ and yet the system be consistent (if the system has superfluous equations; e.g., $x+z=1$, $y=2$, $x+y+z=3$. –  Arturo Magidin Oct 28 '11 at 3:06