How can we derive the standard form of the linear equation: $Ax+By+C$? How can we derive the standard form of the linear equation: $Ax+By+C=0$? What do "$A$", "$B$" and "$C$" in the standard form of the linear equation mean? As in the point-slope form of the linear equation: $y-y_0=m(x-x_0)$, $m$ is the slope, and $x$, $y$ are the coordinates of any point. Likewise, $x_0$, $y_0$ are the coordinates of the given point. 

Let's say we start with the point-slope form $$y-y_0=m(x-x_0),$$ where $(x, y)$ represents any point on the line, and $(x_0, y_0)$ represents a given point on the line, and $m$ is the given slope of the line. Now, 
$$
\begin{align} y-y_0 =m(x-x_0) 
& \iff  y - y_0 = mx - mx_0 \\ \\
& \iff y - y_0 - mx - mx_0 = 0 \\ \\
& \iff (-m)x + y + (-y_0 - mx_0) = 0
\end{align}
$$
Now, we can simply let $A = -m,\; B = 1,\; C = (- mx_0 - y_0)$, which means that $$y + (-m)x + (-y_0 - mx_0) = 0 \iff Ax + By + C = 0$$
