Intersection of Two Lines — Cheat sheet

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The idea

The crossing point satisfies both equations. Solve the system; check the answer in both.

Method	Best when
Substitution	one equation is solved for a variable already
Elimination	coefficients cancel on adding or subtracting
Read off	one line is $x = c$ or $y = c$ : substitute at once

Elimination gives	Meaning
a value for $x$ and $y$	one crossing point
$0 = k$ with $k \ne 0$	no solution: parallel lines
$0 = 0$	every point: the same line twice