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I am currently taking MTE507 Algebra II. One of the first topics discussed was Linear Programming. Here's what I had to say about it:
Linear Algebra in general and Linear Programming in particular encompass a wide variety of skills and applications. Indeed, it is a shame that this topic is often relegated to a one-week or two-week unit at the end of most PreCalculus curricula. The fact that this topic is slowly finding its way into the Algebra II syllabus is very encouraging.
Students who study Linear Programming will learn a valuable problem solving technique for maximizing or minimizing objective functions. In addition to learning a new method for problem solving, the student will review many important skills learned earlier in the course. The student will need to review linear equations, points of intersection, solving 2x2 systems of equations as well as graphing inequalities! This is a great opportunity to share the movie "A Beautiful Mind" about John Nash's Theorem which introduces the concept of Linear Programming for the first time. The movie shows how John Nash's discoveries were applied to the field of Econometrics many years later and even earned Dr. Nash a Nobel Prize!
Solving optimization problems in this way can be done in a traditional fashion with just pencil and paper. However, this is a great time to use technology too! The students can solve the systems of equations graphically using a Graphing Calculator. The student may even be introduced to the Matrix Menu of a TI-83/84 to solve these problems via the tools of Linear Algebra. Using technology also makes it easy to entertain "what if scenarios" where one could change a few parameters in the question to see how they affect the solution.
However, the teacher is not bound by the technology of Graphing Calculators in this day and age. Many schools have access to PC Classrooms with Mathematica, Maple or MATLAB software installed whereby these problems may be addressed using the power of a computer. In fact, I have used a free software platform called SAGE, which is much like Mathematica, exactly for this purpose.