Math 1 · Unit 6 · Lesson 6
Systems in the Real World
Two directions of modeling. A story with two facts about two unknowns, such as a total count and a total value, becomes two equations: solve by the method that fits and read the answer back into the story. A story with two limits, such as a budget and a time limit, becomes two inequalities: the feasible region, the overlap, is every plan that obeys both. Whole things make the plans a set of dots.
The big idea
A story with two unknowns is a system. Two facts, usually a count and a value, give two equations, and the pair that makes both true is the answer: name the unknowns, write the two equations, pick the method that fits, solve, and read the numbers back into the story with a check. Two limits, such as a budget and a time limit, give two inequalities, and the overlap of their regions, the feasible region, is every plan that obeys both at once. When the things are whole, only the whole-number points in the overlap are real plans, so find one and say what it costs and how long it takes. It has to satisfy every equation, or obey every limit, or it is not a solution: the thread of the whole unit.
The parts of this lesson
Each part is its own short page. Go in order, or pick the one you need.
Printables
One page with the count-and-value setup, the five steps, the two-limit setup with the feasible region, discrete plans, and worked examples. Plus a worksheet with a grid and an answer key, and a short lesson plan.