Havlishkq

2022-02-17

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:
$y-{y}_{0}=m\left(x-{x}_{0}\right)$, 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.

Nicolle Newman

$y-{y}_{0}=m\left(x-{x}_{0}\right)$,
where (x,y) represents any point on the line, and $\left({x}_{0},{y}_{0}\right)$ represents a given point on the line, and m is the given slope of the line. Now,
$y-{y}_{0}=m\left(x-{x}_{0}\right)⇔y-{y}_{0}=mx-m{x}_{0}$
$⇔y-{y}_{0}-mx-m{x}_{0}=0$
$⇔\left(-m\right)x+y+\left(-{y}_{0}-m{x}_{0}\right)=0$
Now, we can simply let $A=-m,B=1,C=\left(-m{x}_{0}-{y}_{0}\right)$, which means that
$y+\left(-m\right)x+\left(-{y}_{0}-m{x}_{0}\right)=0\phantom{\rule{0.278em}{0ex}}⟺\phantom{\rule{0.278em}{0ex}}Ax+By+C=0$

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