# Method of determining the correct side

In Linear Programming, we use the “origin side” method to determine which side is the “>” side of the straight line $$f(x, y) = 0$$.

I know it is NOT necessary true that $$f(x, y) > 0$$ is always on the “right hand side” of $$f(x, y) = 0$$. But it seems if we re-write $$f(x, y) = 0$$ with $$((x)) > 0$$, that guess is true.

Question:- Is it the case? If it is a yes, then is it a standard method same as the “origin side” and how can we convince others?

## 1 Answer

The cases of identifying the regions separated by the lines (y = constant or x = constant) are too trivial to consider. Hence, we can assume the slanted line is $$L : y = mx + c$$ with $$m \ne 0$$.

Re-write the equation as $$L : x = \dfrac {1}{m}y + d$$; where $$d = \dfrac {c}{m}$$.

Let D be another constant such that $$D \gt d$$. Then, the point $$(D, 0)$$ is NOT on $$L$$ and in fact it is on the right hand side of $$L$$.

After putting $$x = D$$ and $$y = 0$$ in $$L$$, we can say that the points lying on the same side as $$(D, 0)$$ (i.e. the right hand side of $$L$$) is represented by $$x \gt \dfrac {1}{m}y + d$$.