System trichotomy
SAT glossary · Course Foundations
Definition
The geometric image of the one-variable trichotomy: two lines cross once (slopes differ, a_1/a_2 ≠ b_1/b_2), are parallel and distinct (no solution, a_1/a_2 = b_1/b_2 ≠ c_1/c_2), or are the same line twice (infinitely many, all three ratios equal). There are no other possibilities.
System trichotomy in plain English
When you have two linear equations (a 'system'), there are exactly three geometric possibilities for how their graphs relate, and no others: the two lines cross at exactly one point (their slopes are different, meaning a_1/a_2neqb_1/b_2, giving exactly one solution); the lines are parallel and never touch (same slope but different position, meaning a_1/a_2=b_1/b_2neqc_1/c_2, giving NO solutions); or the two equations actually describe the exact same line drawn twice (all three ratios equal each other, giving INFINITELY many solutions, since every point on the line works). Those three cases are the complete list — there's nothing else that can happen with two straight lines.
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