Warm-up 1.2 Find matrix G if G = [0 2] [0 -3] [-1 0] [0 2] [0 -3] [-1 0] [1 3] + [-1 4] - [2 -2] [2 1] [2 2] [-3 1]

Reminders: Sorry! I’ll be absent tomorrow for a training I have to go to. I will leave specific sub plans. All work must be turned in (I will come by the school and pick up work to grade it). If you have any questions, comment on the website or leave a note on your paper *What level are you?

Topic: Linear Systems & Matrices Key Learning(s): Matrices are a tool for organizing data so that it is easily manipulated. They can be used to obtain additional information and to draw conclusions about the data. Unit Essential Question (UEQ): How can matrices be used to solve a system of equations and inequalities?

Concept I: Solving Systems in Two Variables Lesson Essential Question (LEQ): How do you solve systems graphically and algebraically? Vocabulary: System of Equations Consistent Independent Dependent

Concept II: Solving Systems in Three Variables Lesson Essential Question (LEQ): How do you determine corresponding elements in a matrix? Vocabulary: Ordinary Triple

Concept III: Operations with Matrices Lesson Essential Question (LEQ): How do you model data using matrices? How do you add, subtract, and multiply matrices? Vocabulary: Matrix Elements Dimensions

Concept IV: Determinants and Multiplicative Inverses of Matrices Lesson Essential Question (LEQ): How do you evaluate determinants? How do you find the inverse of a matrix? How do you solve systems of equations by using inverse matrices? Vocabulary: Determinant Minor Identity matrix

Concept V: Solving Systems of Linear Inequalities Lesson Essential Question (LEQ): How do you graph systems of inequalities? How do you find the maximum or minimum value of a function defined for a polygonal convex set? Vocabulary: Polygonal convex set

Lesson Essential Question (LEQ): Concept VI: Linear Programming Lesson Essential Question (LEQ): How do you use systems of linear inequalities to solve real-world problems? Vocabulary: Linear programming Constraints Infeasible Unbounded

§1.2: Organizing Data & Matrices LEQ: How can statistical data be organized into matrices? Uses for matrices… Answers on p. 354 - 355

Topics Organizing Statistical Data What is the difference between a 2x3 matrix and a 3x2 matrix? Presenting Data from a table in a matrix Elements, rows, columns (row, column) Equal matrices = iff same dimensions and corresponding elements are equal Solving matrices with algebra involved

Adding and Subtracting Matrices LEQ: How do you add, subtract, and multiply matrices? Matrix addition/subtraction – add/subtract corresponding elements Explain why you add matrices only if they have the same dimensions. Is matrix subtraction commutative? Why? Matrix equation – addition/subtraction properties of equality

10.3: Matrix Multiplication How are the dimensions of a matrix related to the ability to multiply matrices? Scalar Multiplication (Scalar) You can also multiply two matrices. Method: Multiply the elements of each row of the first matrix by the elements of the first column of the second matrix. Add the products.

Use scalar Multiplication

Assignment Practice 3.1 – 3.2 Mixed Exercises Selected Problems?

Warm-up 1/11/08 Find the product, if possible. If not, possible, write produce undefined. [2] [3 1 -2] [1] [1] [1 0] [2] [0 1] [-1 0 -2] [1] [2 3 1] [3] [2] [6 2 -4] [-3 -1 2] Product undefined [-5] [13]

Identity and Inverse Matrices LEQ: How would you describe the identity matrix and its uses? A square matrix is a matrix with the same number of columns as it has rows. The Identity matrix is a square matrix with 1’s along the diagonal and 0’s everywhere else.

Inverse Matrix IF X is the inverse matrix of A, then AX = I (A times its inverse = the identity) Not all matrices have inverses. If detA = 0, then A does not have an inverse. If detA ≠ 0, then A does have an inverse.

If A = [ a b] [ c d] Then, A-1 = *swap a & c, make b,d neg. 1 [c -b] (ad – bc) [a -d] You can also use inverse matrices to solve matrix equations. [0 -4] X = [0] [0 -1] [4] For this problem A-1 does not exist, so you cannot solve the problem.

Examples 3-6 Worksheet

Solving Systems of Linear Equations in Three Variables Using the Elimination Method Note that there is more than one way that you can solve this type of system.  Elimination (or addition) method is one of the more common ways of solving the problem by algebra, so I choose to show it this way.

Step 1: Simplify and put all three equations in the form Ax + By  + Cz = D if needed . This would involve things like removing ( ) and  removing fractions. To remove ( ): just use the distributive property.  To remove fractions: since fractions are another way to write division, and the inverse of divide is to multiply, you remove fractions by multiplying both sides by the LCD of all of  your  fractions.

 Step 2: Choose to eliminate any one of the variables from any pair of equations. This works in the same manner as eliminating a variable with two linear equations and two variables At this point, you are only working with two of your equations.  In the next step you will incorporate the third equation into the mix. 

Looking ahead, you will be adding these two equations together. Make sure one of the variables cancels out. It doesn't matter which variable you choose to drop out.  For example, if you had a 2x in one equation and a 3x in another equation, you could multiply the first equation by 3 and get 6x and the second equation by -2 to get a -6x.  So when you go to add these two together they will cancel out.

Step 3:  Eliminate the SAME variable chosen in step 2 from any other pair of equations, creating a system of two equations and 2 unknowns. Basically, you are going to do another elimination step, eliminating the same variable we did in step 2, just with a different pair of equations.  Follow the same basic logic as shown in step 2

Step 4:  Solve the remaining system found in step 2 and 3, just as if it is a system of 2 equations in 2 variables  After steps 2 and 3, there will be two equations and two unknowns which is just a system of 2 equations and 2 variables. You can use any method you want to solve it. When you solve this system that has two equations and two variables, you will have the values for two of your variables.

Remember that if both variables drop out and you have a FALSE statement, that means your answer is no solution.  If both variables drop out and you have a TRUE statement, that means your answer is infinite solutions, which would be the equation of the line.

Step 5:  Solve for the third variable. If you come up with a value for the two variables in step 4, that means the three equations have one solution.  Plug the values found in step 4 into any of the equations in the problem that have the missing variable in it and solve for the third variable.

Step 6:  Check. You can plug in the proposed solution into ALL THREE equations.  If it makes ALL THREE equations true then you have your solution to the system.  If it makes at least one of them false, you need to go back and redo the problem.

There are three possible outcomes that you may encounter when working with these systems: one solution no solution infinite solutions

Solve for x, y, and z -3x + 8y + 1z = -18 -2x – 6y – 9z = -15 -6x + 5y + 3z = 13 (-3,-4,5)

Solving systems using Matrices Ex. [A]x = [C] Typically, you would divide both sides by [A] Division with matrices is notated as [A]-1. So, [A]-1[A] = I Do the same thing to the other side [A]-1[C] Remember multiplication is not commutative, thus, you always do the order the same way!

Assignment #1-3 on back of WS Complete Worksheet (you may use calculators) HW: Take home quiz

Graph each of these inequalities: Warm-up 1/14/08 Graph each of these inequalities: 2x – 3y < 12 x + 5y < 20

Solving Systems of Linear Inequalities: LEQ: How do you graph systems of inequalities? Graph equations Shade region

Ex) y > ( 2/3)x – 4 y < (– 1/5)x + 4 x > 0 The solution is where all the inequalities work (the region where all three individual solution regions overlap) In this case, the solution is the shaded part in the middle

Some systems may have regions that are completely bounded Other systems will have regions that are unbounded in one direction Still other systems will have no common solution region

LEQ: How do you find the maximum or minimum value of a function defined for a polygonal convex set? Linear programming is the process of taking various linear inequalities relating to some situation, and finding the "best" value obtainable under those conditions.

In "real life", linear programming is part of a very important area of mathematics called "optimization techniques". This field of study (or at least the applied results of it) are used every day in the organization and allocation of resources. These "real life" systems can have dozens or hundreds of variables, or more. In algebra, though, you'll only work with the simple (and graphable) two-variable linear case.

The general process is to graph the inequalities (which, in this context, are called "constraints") Form a walled-off area on the x,y-plane (which is called the "feasibility region"). Figure out the coordinates of the corners of this feasibility region (that is, the intersection points of the various lines), Test these corner points in the formula (called the "optimization equation") for which you're trying to find the highest or lowest value.

Find the maximal and minimal value of z = 3x + 4y subject to the following constraints:

To solve Graph the system Find the intersection points The maximum and minimum will be at the corner points Plug in the important points you found to find the minimum and maximum values

Assignment Graphing Systems of Inequalities Finding minimum and maximum values of a function.

Warm-up 1/15/08 Solve each system of equations or inequalities. x + 2y = 8 -x – 3y = 2 9x + 12y < 36 34x – 17y > 34

Given the inequalities, linear-programming exercises are pretty straightforward, if sometimes a bit long. The hard part is usually the word problems, where you have to figure out what the inequalities are. So I'll show how to set up some typical linear-programming word problems.

At a certain refinery, the refining process requires the production of at least two gallons of gasoline for each gallon of fuel oil. To meet the anticipated demands of winter, at least three million gallons a day will need to be produced. The demand for gasoline, on the other hand, is not more than 6.4 million gallons a day. If gasoline is selling for $1.90 per gallon and fuel oil sells for $1.50/gal, how much of each should be produced in order to maximize revenue?

The question asks for the number of gallons which should be produced, so I should let my variables stand for "gallons produced". x: gallons of gasoline produced y: gallons of fuel oil produced

Since this is a "real world" problem, I know that I can't have negative production levels, so the variables can't be negative. This gives me my first two constraints: namely, x > 0 and y > 0.

Since I have to have at least two gallons of gas for every gallon of oil, then x > 2y. For graphing, of course, I'll use the more manageable form "y < ( 1/2 )x".

The winter demand says that y > 3,000,000; note that this constraint the eliminates the need for the "y > 0" constraint. The gas demand says that x < 6,400,000.

I need to maximize revenue R, so the optimization equation is R = 1 I need to maximize revenue R, so the optimization equation is R = 1.9x + 1.5y. Then the model for this word problem is as follows: R = 1.9x + 1.5y , subject to: x > 0 x < 6,400,000    y > 3,000,000 y < ( 1/2 )x

When you test the corner points at (6. 4m, 3. 2m), (6 When you test the corner points at (6.4m, 3.2m), (6.4m, 3m), and (6m, 3m), you should get a maximal solution of R = $16.96m at (x, y) = (6.4m, 3.2m).

10% optimization/application problems Test Expectations: Fewer than 20 problems 30% word problems 60% regular problems 10% optimization/application problems

Warm-up 1/16/08 Create a system of 3 equations to solve the problem: James sold magazine subscriptions with three prices: $20, $11, and $28. He sold 3 fewer of the $20 subscriptions than of the $11 subscriptions and sold a total of 37 subscriptions. If his total sales amounted to $735, how many $28 subscriptions did James sell?

Solve: Tasty Bakery sells three kinds of muffins: chocolate chip muffins at 25 cents each, oatmeal muffins at 30 cents each, and blueberry muffins at 35 cents each. Charles buys some of each kind and chooses twice as many blueberry muffins as chocolate chip muffins. If he spends $4.35 on 14 muffins, how many of each did he buy?