Solution trichotomy
SAT glossary · Course Foundations
Definition
The complete classification of ax+b=cx+d via (a-c)x = d-b: exactly one solution when a ≠ c; no solution (contradiction) when a=c, b≠ d; infinitely many (identity) when a=c, b=d. The examination tests all three branches, most often in reverse — telling you the branch and asking for the parameter.
Solution trichotomy in plain English
For a linear equation like ax+b=cx+d, there are exactly three possible outcomes once you simplify it down to (a-c)x=d-b, and this covers every single case with nothing left over: if a≠ c, there's exactly ONE solution; if a=c but b≠ d, you get a contradiction (something like 0=7), meaning NO solution exists; and if a=c AND b=d, the equation becomes an identity (something like 0=0) that's true for literally every value of x, giving INFINITELY many solutions. The test often tests this backwards — telling you which of the three outcomes it wants and asking you to find what constant would produce it.
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