Why does the sign of the result determine the location of the point?

Question

I had learned this in school:- if we have an equation of a line (say x+y-6=0) if the result from substituting a point comes out to be:

1. negative = then the point is above the line
2. positive = then the point is below the line

So If I put in the point (3,2), then it would be 3+2-6 = -1 = negative = point above the line.

I attempted to prove it but wasn't able to think of any theorems/logic that can aid this.

Can anyone explain the proof or any logic that explains why this works?? Or is this determined experimentally?

Example:

Answer 1

This is easily understood from a continuity argument.

The line is the locus of the points of coordinates such that $f(x,y)=ax+by+c=0$. By continuity (Intermediate Value theorem), you cannot move from a point where $f$ is positive to a point where it is negative without crossing the line. Hence the line splits the plane by sign.

(For a more correct description, you should replace "above" and "below" by "left" and "right", where the orientation depends on the signs of the coefficients.)

Answer 2

This is not true.

The above graph shows the lines $x+y-6=0$ and $-x-y-6=0$. If we put in $(3,2)$ into each of the lines, we get $$3+2-6=-1<0$$ and $$-3-2-6=-11<0$$ respectively. Both give a negative result, but the point is below the first line and above the second.