Ch 2. 6 – Solving Systems of Linear Inequalities & Ch 2 Ch 2.6 – Solving Systems of Linear Inequalities & Ch 2.7 – Linear Programming

Vocabulary (2.6) Polygonal Convex Set: when 2 or more linear inequalities make a bounded set of points.

Function Notation You are familiar with f(x) =ax + b, where a and b are real numbers Functions may be defined with more than one variable: Ex) f(x, y) = ax + by + c where a,b,c are real numbers

Vertex Theorem The minimum or maximum of f(x, y) = ax + by + c on a polygonal convex set occurs at a vertex of the polygonal boundary.

Example

Vocabulary (2.7) Linear Programming: using polygonal convex sets to find maximums or minimums of real world problems Constraints: conditions of the real life problem. The inequality systems that creates the polygonal convex set.

Linear Programming Procedure Define variables Write constraints Graph and find vertices of polygonal convex set Write an expression whose value is to be maximized or minimized Substitute values of the vertices in the expression Select the greatest or least value

The profit on each set of cassettes that is manufactured by MusicMan, Inc., is $8. The profit on a single cassette is $2. Machines A and B are used to produce both types of cassettes. Each set takes nine minutes on Machine A and three minutes on Machine B. Each single takes one minute on Machine A and one minute on Machine B. If Machine A is run for 54 minutes and Machine B is run for 42 minutes, determine the combination of cassettes that can be manufactured during the time period that most effectively generates profit within the given constraints.

Sometimes there might not be a polygonal convex set Infeasible: when the constraints cannot be satisfied simultaneously. (no solution) Unbounded: the region formed by the constraints is not a closed region. (just a minimum or just a maximum value)

More than one solution? Alternate Optimal Solutions: when two or more vertices of a polygonal convex set gives the same minimum or maximum