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					                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

				
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