Unit 3

Turn in your interimsUnit 3Linear ProgrammingSolving Systems of Equations with 3 VariablesInverses & Determinants of Matrices Cramers RuleLinear ProgrammingWhat is it?Technique that identifies the minimum or maximum value of a quantityObjective functionLike the parent functionConstrains (restrictions)Limits on the variablesWritten as inequalitiesWhat is the name of the region where our possible solutions lie?Feasible regionContains all of the points which satisfy the constraintsVertex Principle of Linear ProgrammingIf there is a max or a min value of the linear objective function, it occurs at one or more vertices of the feasible regionTesting VerticesFind the values of x and y that maximize and minimize P?

What is the value of P at each vertex?

1. Graph the constraints

2. Find coordinates of each vertex3. Evaluate P at each vertex

when x=4 and y=3 P has a max value of 18

Furniture ManufacturingA furniture manufacturer can make from 30 to 60 tables a day and from 40 to 100 chairs a day. It can make at most 120 units in one day. The profit on a table is $150, and the profit on a chair is $65. How many tables and chairs should they make per day to maximize profit? How much is the maximum profit?Define our variables:X: number of tables Y: number of chairsx+y Choose the matrix

Verifying InversesMultiply the matrices to ensure result is IIf not then the two matrices are not inverses

A=B=

AB= =

AB=Solving a System of Equations with Matrices

(4, -10, 1)Practice Problems

2.

3.

(5,-3)(5,0,1)(1,0,3)Practice Solving Systems with MatricesSuppose you want to fill nine 1-lb tins with a snack mix. You plan to buy almonds for $2.45/lb, peanuts for $1.85/lb, and raisins for $.80/lb. You have $15 and want the mix to contain twice as much of the nuts as of the raisins by weight. How much of each ingredient should you buy?

Let x represent almondsLet y represent peanutsLet z represent raisins

33Calculator How To!!To input a matrix:2nd, Matrix, EditBe sure to define the size of your matrix!!To find the inverse of a matrix2nd, Matrix, 1, x-1, enterHomeworkP. 50 # 1, 2, 6, 9, 10, 11, 13, 14