Coincident Lines — Cheat sheet

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The condition

a1a2=b1b2=c1c2\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} - all three ratios agree, so one equation is a multiple of the other.

Fast test

Divide the first coefficients for the multiplier, then check it carries the other two terms. Failing on the constant alone means parallel.

In a system

Infinitely many solutions means coincident: the second equation adds no information, so no unique point can be found.