A System from a Story

Building a system from a story means naming two unknowns, writing one equation per stated fact, solving by either method, and reading the pair back into the story's units.

A lab needs 10 liters of 30% acid, mixing a 20% solution with a 60% one. Name the unknowns: x liters of the weak, y of the strong. Fact one, volume: x + y = 10. Fact two, acid content: 0.2x + 0.6y = 3, since 30% of 10 liters is 3 liters of pure acid. Substitute y = 10 − x: 0.2x + 6 − 0.6x = 3, so −0.4x = −3 and x = 7.5, giving y = 2.5. The audit in units: 0.2(7.5) + 0.6(2.5) = 1.5 + 1.5 = 3 liters of acid — the blend hits 30% on the nose. One equation per fact is the discipline; the algebra is Unit 3's.