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7.1
Solving Systems of Two Equations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
What you’ll learn about


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The Method of Substitution
Solving Systems Graphically
The Method of Elimination
Applications
… and why
Many applications in business and science can be
modeled using systems of equations.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 2
Solution of a System
A solution of a system of two equations in two
variables is an ordered pair of real numbers that
is a solution of each equation. A system is
solved when all of its solutions are found.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 3
Example Using the Substitution Method
Solve the system using the substitution method.
2 x  y  10
6x  4 y  1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example Solving a Nonlinear System by
Substitution
Solve the system by substitution.
x  y  38
xy  361
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 5
Example Solving a Nonlinear System
Algebraically
Solve the system algebraically.
y  x  6x
y  8x
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 6
Example Solving a Nonlinear System
Graphically
Solve the system:
ln y  x
y  x2  8x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 7
Example Using the Elimination Method
Solve the system using the elimination method.
5 x  3 y  21
3x  2 y  5
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 8
Example Finding No Solution
Solve the system:
3x  2 y  5
6 x  4 y  10
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 9
Example Finding Infinitely Many
Solutions
Solve the system.
3x  6 y  10
9 x  18 y  30
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 10
Example Solving Word Problems with
Systems
Find the dimensions of a rectangular cornfield with a perimeter of
220 yd and an area of 3000 yd2.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 11
Homework


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Homework Assignment #9
Read Section 7.2
Page 575, Exercises: 1 – 65 (EOO)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 12
7.2
Matrix Algebra
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
The points (a) (1,  3) and (b) ( x, y) are reflected across the given line.
Find the coordinates of the reflected points.
1. The x-axis
2. The line y  x
3. The line y   x
Expand the expression,
4. sin( x  y )
5. cos( x  y )
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 14
Quick Review Solutions
The points (a) (1,  3) and (b) ( x, y ) are reflected across the given line.
Find the coordinates of the reflected points.
1. The x-axis (a) (1,3) (b) ( x,  y )
2. The line y  x (a) (  3,1) (b) ( y, x)
3. The line y   x (a) (  3,  1) (b) ( y,  x)
Expand the expression,
4. sin( x  y ) sin x cos y  sin y cos x
5. cos( x  y ) cos x cos y  sin x sin y
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 15
What you’ll learn about
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Matrices
Matrix Addition and Subtraction
Matrix Multiplication
Identity and Inverse Matrices
Determinant of a Square Matrix
Applications
… and why
Matrix algebra provides a powerful technique to manipulate large
data sets and solve the related problems that are modeled by the
matrices.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 16
Matrix
Let m and n be positive integers. An m × n matrix
(read "m by n matrix") is a rectangular array of m
rows and n columns of real numbers.
 a11
a
 21


 am1
a12
a22
am 2
a1n 
a2 n 



amn 
We also use the shorthand notation  aij  for this matrix.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 17
Matrix Vocabulary
Each element, or entry, aij, of the matrix uses
double subscript notation. The row subscript is
the first subscript i, and the column subscript is
j. The element aij is in the ith row and the jth
column. In general, the order of an m × n
matrix is m×n.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 18
Example Determining the Order of a
Matrix
What is the order of the following matrix?
1 4 5
3 5 6 


Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 19
Matrix Addition and Matrix Subtraction
Let A   aij  and B  bij  both be matrices of order m  n.
1. The sum A + B is the m  n matrix A  B   aij  bij  .
2. The difference A  B is the m  n matrix A  B   aij  bij  .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 20
Example Matrix Addition
1 2 3  2 3 4 
4 5 6  5 6 7 

 

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 21
Example Using Scalar Multiplication
1 2 3
3

4
5
6


Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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The Zero Matrix
The m  n matrix 0  [0] consisting entirely of zeros is
the zero matrix.
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Additive Inverse
Let A   aij  be any m  n matrix.
The m  n matrix B   aij  consisting of the additive
inverses of the entries of A is the additive inverse
of A because A  B  0.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 24
Matrix Multiplication
Let A   aij  be any m  r matrix and B  bij  be any r  n matrix.
The product AB  cij  is the m  n matrix where
cij  ai1b1 j +ai 2b2 j  ...  air brj .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 25
Example Matrix Multiplication
Find the product AB if possible.
1 2 3 
A

0
1

1


1 0 
and B   2 1 


 0 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 26
Identity Matrix
The n  n matrix I n with 1's on the main diagonal and 0's elsewhere
is the identity matrix of order n  n.
1
0

I n  0


0
0 0
1 0
0 1
0 0
0
0
0

0

0
1 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 27
Inverse of a Square Matrix
Let A   aij  be an n  n matrix. If there is a matrix B
such that AB  BA  I n , then B is the inverse of A.
We write B  A1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 28
Inverse of a 2 × 2 Matrix
1
a b 
1  d b 
If ad  bc  0, then 

.



ad  bc  c a 
c d 
The number ad  bc is the determinant of the 2  2 matrix
a b
a b 
A
and is denoted det A 
 ad  cb.

c d
c d 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 29
Minors and Cofactors of an n × n Matrix
If A is an n  n matrix where n  2, the minor M ij corresponding
to the element aij is the determinant of the  n  1   n  1 matrix
obtained by deleting the row and column containing aij .
The cofactor corresponding to aij is Aij   1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
i j
M ij .
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Determinant of a Square Matrix
Let A   aij  be a matrix of order n  n (n  2). The
determinant of A, denoted by det A or | A | , is the
sum of the entries in any row or any column multiplied
by their respective cofactors. For example, expanding
by the i th row gives det A | A | ai1 Ai1  ai 2 Ai 2  ...  ain Ain .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 31
Transpose of a Matrix
Let A   aij  be a matrix of order n  m. The transpose
of A, denoted by AT is the matrix in which the rows in
A become the columns in AT and the columns in A
become the rows in AT or AT   a ji  .
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 32
Example Using the Transpose of a Matrix
If pizza sizes are given by the matrix
Size   Pers Sm Med Larg  , pizza sales are given by
the matrix Sales  55 25 15 10 , and pizza prices are
given by the matrix Price  $2.50 $3.50 $7.50 $11.50 ,
what are the total sales for the day?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 33
Inverses of n × n Matrices
An n × n matrix A has an inverse if and only if
det A ≠ 0.
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Slide 7- 34
Example Finding Inverse Matrices
 1 3
Find the inverse matrix if possible. A  

2
5


Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 35
Properties of Matrices
Let A, B, and C be matrices whose orders are such that the following sums,
differences, and products are defined.
1. Community property
Addition: A + B = B + A
Multiplication: Does not hold in general
2. Associative property
Addition: (A + B) + C = A + (B + C)
Multiplication: (AB)C = A(BC)
3. Identity property
Addition: A + 0 = A
Multiplication: A·In = In·A = A
4. Inverse property
Addition: A + (-A) = 0
Multiplication: AA-1 = A-1A = In |A|≠0
5. Distributive property
Multiplication over addition: A(B + C) = AB + AC (A + B)C = AC + BC
Multiplication over subtraction: A(B - C) = AB - AC (A - B)C = AC - BC
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 36
Reflecting Points About a Coordinate
Axis
To reflect a point about the x-axis, express the point as a
1 0 
1 2 matrix and multiply by 
to obtain the 1 2

0 1
matrix of the reflected point.
To reflect a point about the y -axis, express the point as a
 1 0 
1 2 matrix and multiply by 
to obtain the 1 2

 0 1
matrix of the reflected point.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 37
Example Using Matrix Multiplication
46. A company has two factories, each manufacturing three products. The number
of products i made in factory j in one week is given by aij in the matrix
120 70 
A  150 110  . If production is increased by 10%, write the new production levels


 80 160 
as a matrix B. How is B related to A?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 38