= Solution
Set $\Delta x=x_B-x_A$ and $\Delta y=y_B-y_A$. A displacement from $A$ lies along the <straight line> precisely when its two-dimensional <determinant> with $(\Delta x,\Delta y)$ vanishes. Thus an <implicit equation> is
$$
\boxed{\Delta y(x-x_A)-\Delta x(y-y_A)=0.}
$$
Equivalently, write $ax+by+c=0$, with $a=\Delta y$, $b=-\Delta x$ and $c=\Delta x\,y_A-\Delta y\,x_A$. The <vector> $(a,b)$ is a <normal vector>. The formula handles vertical and horizontal <straight lines> without division. It presupposes $A\ne B$: coincident <points> do not determine a unique <straight line>.
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