Pick one point on the plane, say $\displaystyle (x_{1},y_{1})$ and other one$\displaystyle (x,y)$. We will try to see the relation between $\displaystyle x$ and $\displaystyle y$ Now impose that they are in a line:
Let $\displaystyle \alpha$ be the angle between the black and the red segments. We have $\displaystyle \tan(\alpha)=\frac{\Delta y}{\Delta x}$ where $\displaystyle \Delta y=y-y_1$ and $\displaystyle \Delta x=x-x_1$. Then
$\displaystyle \tan(\alpha)(x-x_1)=(y-y_1) \rightarrow y-y_1=x\tan(\alpha)}-x_{1}\tan (\alpha)$
This is
$\displaystyle \boxed{y=mx+b}$ where $\displaystyle \boxed{m=\tan(\alpha)}$ and $\displaystyle \boxed{b=-x_1 \tan(\alpha)+y_{1}}$