Testing a Point Against a Line — Cheat sheet

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The test

$P(x_0, y_0)$ is on $ax + by + c = 0$ when $ax_0 + by_0 + c = 0$ . Substitute; zero means yes.

Question	Solve for
Is $P$ on the line?	nothing - just evaluate
Missing coordinate of a point	$x$ or $y$
Unknown coefficient in the line	the parameter, $k$

Point	Note
Any $x$ gives a point	choose $x$ , solve for $y$ : that is how lines are tabulated
Result is not zero	the point is off the line; the sign tells you which side
Two points both check out	the line through them is that line