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Learning Target
 LT 2: I can model a real-world scenario
using a system of equations and find the
solution(s).
Modeling with Systems of Equations
 There are 150 adults and 225 children at a zoo. If the zoo makes a
total of $5100 from the entrance fees, and the cost of an adult
and a child to attend is $31, how much does it cost for a parent or
child to attend individually.
Let a be adults
Let c be children
150a + 225c = 5100
How can we improve the definition of
these variables?
A –– What
What isis the
the meaning
meaning of
of the
the
Partner B
Describe
the
meaning
of
the
equation.
term 225c?
150a? What
What are its units?
Modeling with Systems of Equations
 There are 150 adults and 225 children at a zoo. If the zoo makes a
total of $5100 from the entrance fees, and the cost of an adult
and a child to attend is $31, how much does it cost for a parent or
child to attend individually.
Let a be the price of an adult ticket
Let c be the price of a child’s ticket
150a + 225c = 5100
What is another equation that “a” and
“c” must satisfy?
Modeling with Systems of Equations
Fold and label
your paper:
150a + 225c = 5100
a + c = 31
A1)
B1)
A2)
B2)
Partner A solves the system of equations
using ANY METHOD, explaining their work
to Partner B.
Partner B listens and asks questions to
clarify or understand Partner A.
Modeling with Systems of Equations
Fold and label
your paper:
150a + 225c = 5100
a + c = 31
A1)
B1)
A2)
B2)
Partner B solves the system of equations
using a DIFFERENT METHOD, explaining
their work to Partner A.
Partner A listens and asks questions to
clarify or understand Partner B.
Modeling with Systems of Equations
150a + 225c = 5100
a + c = 31
What was effective / ineffective about your solution method?
Which method allowed you to solve the problem more easily?
Why?
Show the solution method that you found more effective on
your worksheet and explain why you chose to solve the system
of equations in that way.
Modeling with Systems of Equations
 Nicole has 15 nickels and dimes. If the value of her coins is $1.20,
how many of each type of coin does she have?
Let …
Let …
n + d = 15
Partner A – Define the variables.
Partner
B – Whatequation
is the meaning
Write a second
that theof the
first
equation
that
has been written?
variables
must
satisfy.
Modeling with Systems of Equations
 Nicole has 15 nickels and dimes. If the value of her coins is $1.20,
how many of each type of coin does she have?
Let n be the number of nickels that Nicole has
Let d be the number of dimes that Nicole has
n + d = 15
What is another way the second
equation can be written?
Modeling with Systems of Equations
Fold and label
your paper:
n + d = 15
5n +10d = 120
A1)
B1)
A2)
B2)
Partner B solves the system of equations
using ANY METHOD, explaining their work
to Partner B.
Partner A listens and asks questions to
clarify or understand Partner B.
Modeling with Systems of Equations
Fold and label
your paper:
n + d = 15
5n +10d = 120
A1)
B1)
A2)
B2)
Partner A solves the system of equations
using a DIFFERENT METHOD, explaining
their work to Partner B.
Partner B listens and asks questions to
clarify or understand Partner A.
Modeling with Systems of Equations
n + d = 15
5n +10d = 120
What was effective / ineffective about your solution method?
Which method allowed you to solve the problem more easily?
Why?
Show the solution method that you found more effective on
your worksheet and explain why you chose to solve the system
of equations in that way.
Modeling with Systems of Equations
Modeling with Systems of Equations
Every group member will solve the problem using a different
method (groups of 4 can have one repeated method)
Learning Log Entries
Write a summary for today’s Learning Target:
 LT 2: I can model a real-world scenario
using a system of equations and find the
solution(s).
Modeling Mixture Problems
How many mL of a 20% acid solution and 12% acid solution
should be mixed to yield 300 mL of a 18% solution?
Today’s Learning Target
 LT 6: I can multiply 2x2 matrices by hand
and larger matrices using technology.
Definition of a Matrix
 A matrix is a rectangular arrangement of numbers in horizontal
rows and vertical columns. The numbers in a matrix are its
elements.
3 columns
2 rows
4  1 5
A

0 6 3
The element in
the first row and
third column is 5
(ar,c).
Definition of a Matrix (Cont’d)
4  1 5
A

0 6 3
3 columns
2 rows
 The dimensions of a matrix with m rows and n columns is m x n
(read “m by n”)
A is a 2x3 matrix.
Definition of a Matrix (Cont’d)
 Two matrices are equal if their dimensions are the same and the
elements in corresponding positions are equal.
 4 0
4  1 5 



1
6
0 6 3 


  5 3


BUT…
1 1
 5  5 
   
 2  2
Quick Check
 Identify the dimensions of each matrix.
2
 1
 
 5 
3 5
1
 5  6 5


1
 3

7

0
4
a 
1

b
 3
 3 3
Quick Check
 Identify the position of the circled element of the matrix (ar,c).
3 5
1
A


5

6
5


1
 3
B
7

0
4
a 
1

b
2
C   1
 5 
D   3  3 3
Multiplying Matrices
 The product of two matrices A and B is defined only if the number
of columns in A is equal to the number of rows in B.
If A is a 4 X 3 matrix and B is a 3 X 5 matrix, then
the product AB is a 4 X 5 matrix.
MULTIPLYING TWO MATRICES
A

B

AB
4 X 3

3 X 5

4X5
4 rows
4 rows
5 columns
5 columns
Multiplying Matrices
 The element in row 1, column 1 of the product of two matrices
can be determined by multiplying row 1 by column 1 and adding
the products:
Multiplying Matrices
 The element in row 1, column 1 of the product of two matrices
can be determined by multiplying row 1 by column 1 and adding
the products:
a b   e
c d    g

 
f  ae  bg


h  ce  dg
af  bh 

cf  dh 
Multiplying Matrices
 Multiply:
1 4  5  7 
3  2  9 6 

 

Multiplying Matrices
 Multiply:
5  7  1 4 
9 6   3  2

 

Multiplying Matrices
 Notice:
5  7  1 4  1 4  5  7 
9 6   3  2  3  2  9 6 

 
 
 

Therefore, matrix multiplication is not commutative.
Multiplying Matrices
 Multiply:
3  2  0 1
1  4   1 3

 

 Now check your answer by using the calculator.
Multiplying Matrices
 Multiply:
3  2 1 0
1  4  0 1

 

 Now check your answer by using the calculator.
Multiplying Matrices
 Use the calculator to multiply:
1  3
4 2   1 0

 0 1 


7  1
The Identity Matrix
The identity matrix is an nxn matrix that has
1’s on the main diagonal and 0’s
elsewhere.
If A is any nxn matrix and I is the nxn
identity matrix, then IA  A .
Multiplying Matrices
Check your answers to HW 4.3
(#2 and #4) using the calculator.
Learning Log Entries
Write a summary for today’s Learning Target:
 LT 6: I can multiply 2x2 matrices by hand
and larger matrices using technology.
Today’s Learning Targets
 LT 3: I can represent a system of
equations using matrices.
 LT 4: I can solve a system of equations
using an inverse matrix.
Multiplying Matrices (Cont’d)
 How can we re-write the following using multiplication?
 8  10  x  24
    
6

5   y  2 

Multiplying Matrices (Cont’d)
 How can we re-write the following based on the definition of
equal matrices.
8 x  10 y  24
 6x  5 y    2 

  
Systems of Equations
 A system of equations of the form:
can be re-written as:
Ax  By  M
Cx  Dy  N
 A B   x  M 
C D    y    N 

    
A X  B
 What needs to happen in order to solve for x and y?
Inverse Matrices
 The inverse of matrix A is A-1 such that
A  A1  I
.
 Find the inverse of A using the calculator and verify that they are
inverses.
3 5
A

1
2


Inverse Matrices
 The inverse of matrix A is A-1 such that
A  A1  I
.
 Find the inverse of A using the calculator and verify that they are
inverses.
2 3 1
A  2 1 0
1 3 1
Systems of Equations
Ex) Solve for the variable matrix using a calculator, showing your work.
 1  1  x   11 
 4 5    y    16

   

Systems of Equations
Ex) Solve for the variable matrix using a calculator, showing your work.
 2 1 3   x   15 
 1 0  1    y    3

    
 0 5  2  z   10 
Learning Log Entries
Write a summary for today’s Learning Target:
 LT 3: I can represent a system of equations
using matrices.
Today’s Learning Target
 LT 5: I can write the inverse matrix A-1 for
a 2x2 matrix A and describe their
product.
The Determinant
Each square matrix (nxn) is associated with a real number called its
determinant. The determinant of matrix A is denoted by det A or A .
The Determinant
 3 2 
a) Find det A if A  


1

4


b) Evaluate
5 4
1 3
Inverse Matrices
The inverse of the matrix
1  d  b
A 
det A  c a 
1
a b 
A

c
d


is
, provided det A  0 .
Inverse Matrices
a) Find
A-1
 3 8
if A  

2
5


6 1 
b) Find the inverse of 

2
4


Learning Log Entries
Write a summary for today’s Learning Target:
 LT 5: I can write the inverse matrix A-1 for a
2x2 matrix A and describe their product.
Today’s Learning Targets
 LT 4: I can solve a system of equations
using an inverse matrix.
Solving a Matrix Equation
Solve the matrix equation for x and y by hand.
  1 2  x   4 
 3 4  y     2

   
Solving a Matrix Equation
Solve the matrix equation for x and y by hand.
 2  7   x   21 3 
 1 4   y    12  2

  

Solving a Matrix Equation
Solve the matrix equation for x and y by hand.
1 1  x   2 3
4 5  y    1 6

  

Systems of Equations
Ex) Write the system of linear equations as a matrix equation and solve
using the inverse matrix.
2 x  3 y  19
x  4 y  7
Systems of Equations
Ex) Write the system of linear equations as a matrix equation and solve
using the inverse matrix.
4 x  y  10
 7 x  2 y  25