Distance between point and line Proof

Question

I want to proof that distance between point and line is equal to
.

So I have next system
,

first part is line formula, and second – formula of perpendicular line from point to this line.
Next we change first expression and get this
$$A (x – x_0) + B (y – y_0) + A x_0 + B y_0 + C = 0 \ .$$

And finally we get this
.

Can someone explain what calculation shall I do to get this expression
$$A (x – x_0) + B (y – y_0) + A x_0 + B y_0 + C = 0 \ .$$

Original source: http://www.mathelp.spb.ru/book1/line_on_plane.htm

Answer 1

We have Ax+ By+ C= 0. Add and subtract $Ax_0$ and $By_0$:

$Ax+ By+ C+ Ax_0- Ax_0+ By_0- By_0= 0$.

$Ax-Ax_0+ By-By_0+ C+ Ax_0+ By_0= 0$

$A(x- x_0)+ B(y- y_0)+ C+ Ax_0+ By_0= 0$.