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Math notes/Pre-Calculus/Analytic Geometry

Relative Positions of Two Lines — Cheat sheet

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The three cases

Ratios	Position	Shared points
$\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}$	intersecting	one
$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}$	parallel	none
$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$	coincident	every point

What solving reveals

The algebra collapses to	Position
$x$ and $y$ values	intersecting
$0 = k$ , $k \ne 0$	parallel
$0 = 0$	coincident

Trap

Compare ratios in a fixed order, always the same line on top. $\dfrac{a_1}{a_2}$ against $\dfrac{b_2}{b_1}$ is the classic slip.