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7.1
Solving Systems of Two Equations
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What you’ll learn about
The Method of Substitution Solving Systems Graphically The Method of Elimination Applications
… and whyMany applications in business and science can be modeled using systems of equations.
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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.
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Example Using the Substitution Method
Solve the system using the substitution method.
2 10
6 4 1
x y
x y
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Example Solving a Nonlinear System by Substitution
Solve the system by substitution.
38
361
x y
xy
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Example Solving a Nonlinear System Algebraically
2
Solve the system algebraically.
6
8
y x x
y x
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Example Solving a Nonlinear System Graphically
2
Solve the system:
ln
8
y x
y x x
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Example Using the Elimination Method
Solve the system using the elimination method.
5 3 21
3 2 5
x y
x y
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Example Finding No Solution
Solve the system:
3 2 5
6 4 10
x y
x y
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Example Finding Infinitely Many Solutions
Solve the system.
3 6 10
9 18 30
x y
x y
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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.
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Homework
Homework Assignment #9 Read Section 7.2 Page 575, Exercises: 1 – 65 (EOO)
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7.2
Matrix Algebra
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Quick Review
The points (a) (1, 3) and (b) ( , ) are reflected across the given line.
Find the coordinates of the reflected points.
1. The -axis
2. The line
3. The line
Expand the expression,
4. sin( )
5. cos
x y
x
y x
y x
x y
( )x y
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Quick Review Solutions
The points (a) (1, 3) and (b) ( , ) are reflected across the given line.
Find the coordinates of the r
(a) (1,3) (b) ( ,
eflected points.
1. The -axis
2. The line
3.
)
(a) ( 3,
Th
1) (b) ( , )
e
x yx
x x
x
y y
y
line
Expand the expression,
4.
(a) ( 3, 1) (b) ( , )
sin cos sin cossin( )
5. cos( cos cos sin s n) i
y x
x y y x
x y x
y
y
x
x y
x y
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What you’ll learn about
Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix Applications
… and whyMatrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices.
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Matrix
11 12 1
21 22 2
1 2
Let and be positive integers. An
(read " by matrix") is a rectangular array of
rows and columns of real numbers.
We also use the short
n
n
m m mn
m n
m n m
n
a a a
a a a
a a a
matrix
m × n
hand notation for this matrix.ija
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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.
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Example Determining the Order of a Matrix
What is the order of the following matrix?
1 4 5
3 5 6
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Matrix Addition and Matrix Subtraction
Let and both be matrices of order .
1. The is the matrix .
2. The is the matrix .
ij ij
ij ij
ij ij
A a B b m n
m n A B a b
m n A B a b
sum +
difference
A B
A B
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Example Matrix Addition
1 2 3 2 3 4
4 5 6 5 6 7
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Example Using Scalar Multiplication
1 2 3
34 5 6
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The Zero Matrix
The matrix 0 [0] consisting entirely of zeros is
the .
m n zero matrix
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Additive Inverse
Let be any matrix.
The matrix consisting of the additive
inverses of the entries of is the
because 0.
ij
ij
A a m n
m n B a
A
A B
additive inverse
of A
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Matrix Multiplication
1 1 2 2
Let be any matrix and be any matrix.
The product is the matrix where
+ ... .
ij ij
ij
ij i j i j ir rj
A a m r B b r n
AB c m n
c a b a b a b
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Example Matrix Multiplication
Find the product if possible.
1 01 2 3
and 2 1 0 1 1
0 1
AB
A B
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Identity Matrix
The matrix with 1's on the main diagonal and 0's elsewhere
is the .
1 0 0 0
0 1 0 0
0 0 1 0
0
0 0 0 0 1
n
n
n n I
I
identity matrix of order
n n
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Inverse of a Square Matrix
1
Let be an matrix. If there is a matrix
such that , then is the of .
We write .
ij
n
A a n n B
AB BA I B A
B A
inverse
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Inverse of a 2 × 2 Matrix
11
If 0, then .
The number is the determinant of the 2 2 matrix
and is denoted det .
a b d bad bc
c d c aad bc
ad bc
a b a bA A ad cb
c d c d
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Minors and Cofactors of an n × n Matrix
If is an matrix where 2, the minor corresponding
to the element is the determinant of the 1 1 matrix
obtained by deleting the row and column containing .
The cofactor correspondin
ij
ij
ij
A n n n M
a n n
a
g to is 1 .i j
ij ij ija A M
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Determinant of a Square Matrix
Let be a matrix of order ( 2). The
determinant of , denoted by det or | | , is the
sum of the entries in any row or any column multiplied
by their respective cofactors. For example, exp
ijA a n n n
A A A
th1 1 2 2
anding
by the row gives det | | ... .i i i i in ini A A a A a A a A
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Transpose of a Matrix
Let be a matrix of order . The transpose
of , denoted by is the matrix in which the rows in
become the columns in and the columns in
become the rows in or .
ij
T
T
T Tji
A a n m
A A
A A A
A A a
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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 sa
les for the day?
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Inverses of n × n Matrices
An n × n matrix A has an inverse if and only if
det A ≠ 0.
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Example Finding Inverse Matrices
1 3Find the inverse matrix if possible.
2 5A
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Properties of MatricesLet A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined.1. Community propertyAddition: A + B = B + AMultiplication: Does not hold in general2. Associative propertyAddition: (A + B) + C = A + (B + C)Multiplication: (AB)C = A(BC)3. Identity propertyAddition: A + 0 = AMultiplication: A·In = In·A = A4. Inverse propertyAddition: A + (-A) = 0Multiplication: AA-1 = A-1A = In |A|≠05. Distributive propertyMultiplication over addition: A(B + C) = AB + AC (A + B)C = AC + BCMultiplication over subtraction: A(B - C) = AB - AC (A - B)C = AC - BC
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Reflecting Points About a Coordinate Axis
To reflect a point about the -axis, express the point as a
1 01 2 matrix and multiply by to obtain the 1 2
0 1
matrix of the reflected point.
To reflect a point about the -axis, express the point
x
y
as a
1 01 2 matrix and multiply by to obtain the 1 2
0 1
matrix of the reflected point.
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Example Using Matrix Multiplication
46. A company has two factories, each manufacturing three products. The number
of products made in factory in one week is given by in the matrix
120 70
150 110 . If production i
80 160
iji j a
A
s increased by 10%, write the new production levels
as a matrix . How is related to ? B B A