# A note on points of intersection

An interesting fact I thought about today. Let us suppose we have to determine the points of intersection of the cartesian equations $f(x,y)=c$ and $g(x,y)=d$. What we generally do is $f(x,y)-c=g(x,y)-d$.

Why? Because the points of intersection $(a,b)$ will, on substitution, make both sides equal. On substitution, the values obtained on both sides will be $0$.

But does the equation $f(x,y)-c=g(x,y)-d$ only determine the points of intersection of the figures $f(x,y)=c$ and $g(x,y)=d$? No. It also determines the points of intersection of the figures $f(x,y)=x+r$ and $g(x,y)=d+r$ for every $r\in\Bbb{R}$.

Shocking, isn’t it? Determining only the points of intersection relevant to $r=0$ can be done by substitution of the points obtained in the two equations separately.