VIEWS: 7 PAGES: 4 POSTED ON: 12/15/2011 Public Domain
Task: Candy? What Candy? Do we get to eat it? (Section 1.11) Suppose you walked into class one day and found a big stack of sealed lunch bags full of candy on a table just waiting for you to rip them open and devour their chocolaty contents. But, you could not even touch them until you figured out how many pieces of each brand of candy was contained in each bag. Well, today is the day. Each group of three gets one bag which must remain unopened until you can tell how many pieces of each type of candy. Each bag holds 3 different types of candy (X, Y, and Z) and a total of 9 pieces of candy. *Your task is to determine exactly what is in your bag by writing a system of equations and solving that system using matrices. BEFORE THE TASK: Review of Matrices Part I-Solving Systems by hand ax by c A system of equations such as can be written as a “matrix equation” where dx ey f a b x c d e y f or AX = B. A is the “coefficient matrix” X is the “variable matrix” B is the “answer matrix” The variable matrix can be isolated by multiplying each side of the equation by the inverse of A: AX = B X = A-1B Example- 4x 2y 30 We will solve the following system of equations: . 3x 5y 19 4 2 x 30 First, write the matrix equation: 3 -5 y 19 1 5 2 30 Next, multiply by the inverse of the constant matrix: 14 3 4 19 x 1 112 8 Then simplify using matrix multiplication: . y 14 14 1 So, x=8 and y=1. Part II-Using Technology (Graphing Calculator) Solving systems of equations of higher order (greater than 2x2) can be accomplished using a similar format and a graphing calculator to find the inverse of the coefficient matrix and find the necessary products. x 2 y 3z 3 To solve the system 2 x y 5z 8 3x y 3z 22 a) Write the coefficient matrix and the answer matrix: A= B= b) Enter A and B into the TI-83/84: 2nd MATRIX EDIT Matrix A (Enter dimensions: 3x3, Enter each entry in the matrix) 2nd MATRIX EDIT Matrix B (Enter dimensions: 3x1, Enter each entry in the matrix) 2nd QUIT, 2nd MATRIX Choose Matrix A, ENTER Use the inverse button X-1 2nd MATRIX Choose Matrix B, ENTER ENTER c) Solution: x= _________ y=_________ z=__________ III: PRACTICE!! 1. For the following systems of equations, write the matrix equation and solve for the variables using your graphing calculator. a. 2x + 3y = 2 b. 9x 7y = 5 c. 5x 4y + 3z = 15 4x 9y = -1 10x + 3y = -16 6x + 2y + 9z = 13 7x + 6y 6z = 6 Part IV-Bag of Candy (Finally) 3. a) Choose values from the Nutrition Chart and the totals given on the card attached to your bag and write a system of equations that describes the information regarding the candy in your group’s paper bag. List your equations below. Equation 1: ________________________________ Equation 2: ________________________________ Equation 3: ________________________________ b) Write the matrix equation that represents this system of equations. c) Solve the matrix equation using a graphing calculator. State the number of pieces of each type of candy. Candy X: _________ Candy Y: _________ Candy Z: _________ Part V: EAT YOUR CANDY!!! Type of Candy Fat Sugar Calories Hershey’s 12g 25g 60g Double Bubble 0g 4g 15g Laffy Taffy 1g 11g 70g Total 38g 113g 380g Type of Candy Fat Sugar Calories Hershey’s 12g 25g 60g Double Bubble 0g 4g 15g Laffy Taffy 1g 11g 70g Total 38g 113g 380g Type of Candy Fat Sugar Calories Hershey’s 12g 25g 60g Double Bubble 0g 4g 15g Laffy Taffy 1g 11g 70g Total 38g 113g 380g Type of Candy Fat Sugar Calories Hershey’s 12g 25g 60g Double Bubble 0g 4g 15g Laffy Taffy 1g 11g 70g Total 38g 113g 380g