Many Unknowns, Counted Ways

Unit 9 finishes the toolkit with two closers — solving several equations at once, and counting without listing — and both run on machinery the course already owns.

A system is a meeting of conditions, and its solution is the point every condition tolerates. The two methods are old laws redeployed: substitution is Unit 5's machine-feeding — an expression fed where a variable sat — and the addition method is the both-sides law scaled and stacked, since multiplying an equation by a constant changes none of its solutions. The trichotomy returns in system clothes: one solution, infinitely many, none — independent, dependent, inconsistent — with parallel lines playing the role 3 = 5 played in Unit 3. Three variables raise the ladder without changing it: eliminate down to two, then one, then climb back up, and the mixture problems of the world are systems wearing lab coats — one equation per stated fact, units audited at the end, 7.5 liters of the weak and 2.5 of the strong landing the blend at 30% exactly. Counting then replaces listing. The Multiplication Principle multiplies choices, factorials count orderings, permutations rank and combinations pool — the 20 ordered pairs collapsing to 10 once order stops mattering. And the binomial coefficients close the course's oldest account: (a + b)ⁿ expands by Pascal's row with every middle term present, which makes the Binomial Theorem the missing-middle error's final defeat — the 2ab of Unit 2's dream was the n = 2 row all along, and 27 = 27 is the receipt at any power.