obj. 24 systems of linear equations (presentation)
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8/2/2019 Obj. 24 Systems of Linear Equations (Presentation)
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Obj. 24 Systems of Linear
Equations
Unit 6 Systems of Equations and Inequalities
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Concepts and Objectives
Systems and Matrices (Obj. #24)
Set up an augmented matrix from a system ofequations
Solve an augmented matrix by calculator
Calculate the determinant of a square matrix Use Cramers Rule to solve a system of equations
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Systems of Linear Equations
A set of equations is called asystem of equations. If all of
the variables in all of the equations are of degree one,then the system is a linearsystem. There are three
possibilities:
There is a single solution that satifies all the
equations.
There is no single solution that satisfies all the
equations.
There are infinitely many solutions to the equations.
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Systems of Linear Equations
There are four different methods of solving a linear
system of equations:1. Substitution Solve one equation for one variable,
and substitute it into the other equation(s).
2. Elimination Transform the equations such that if
you add them together, one of the variables is
eliminated.
3. Graphing Graph the equations, and the solution is
their intersection.4. Matrices Convert the system into one or two
matrices and solve.
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Systems and Matrices
A matrixis a rectangular array of numbers enclosed in
brackets. Each number is called an elementof thematrix.
There are three different ways of using matrices to solve
a system:
Use the multiplicative inverse.
The Gauss-Jordan Method, which uses augmented
matrices.
Cramers Rule, which uses determinants.
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Augmented Matrices
In an augmentedmatrix, the coefficients and constants of
equations in standard form are combined into onematrix.
is written as
While we can solve this system manually, the calculator
makes this much easier, using a process called Reduced
Row-Echelon Form.
+ + =
+ =
+ + =
3 2 1
2 2
2
x y z
x y z
x y z
1 3 2 1
2 1 1 2
1 1 1 2
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Augmented Matrices
Example: Solve, using an augmented matrix.
=+ =
3 4 1
5 2 19
x y
x y
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Augmented Matrices
Solving augmented matrices on the NSpire is
ridiculously easy:1. Press theb key.
2. Select Matrix & Vector
3. Select Reduced Row-Echelon Form4. Press thet key.
5. Select the nn matrix
6. Change the rows to 2
7. Enter the matrix values and press.
Note: you can also just type in rref( and skip 1-3
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Augmented Matrices
Example: Write the system of equations associated with
the augmented matrix. Do not solve.
1 3 6 7
2 1 1 1
1 2 2 1
+ =
+ = + + =
3 6 7
2 1
2 2 1
x y z
x y z
x y z
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Determinants
Every n n matrixA is associated with a real number
called the determinantofA, written |A |. The determinant is the sum of the diagonals in one
direction minus the sum of the diagonals in the other
direction.
Example:
3 4
6 8= = 24 24 48( )( ) ( )( )= 3 8 6 4
a b
c dad cb=
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Determinants
Example: Find the determinant of
2 2
3 1
( )( ) ( )( )
=
2 2
2 1 3 23 1
= + =2 6 8
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Determinants
Example: Solve forx:
=3
4x
x x
=
2
3 4x x =
23 4 0x x
( )( ) + =4 1 0x x
=
4, 1x
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Determinants
To calculate the determinant of a 33 matrix, repeat the
first two columns to help you draw the diagonals:
Again, your calculator can also calculate the determinant
of a matrix you have entered. (Look for det on the
TI-83/84.)
8 2 4
7 0 3
5 1 2
= 7 0
5
8 2
1
4
3
8 2
7
2 5
0
1
= 500= ( )30+ 28+ (0 ( )24+ ( ))28+
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Cramers Rule To solve a system using Cramers Rule, set up a matrix of
the coefficients and calculate the determinant (D). Then, replace the first column of the matrix with the
constants and calculate that determinant (Dx).
Continue, replacing the column of the variable with the
constants and calculating the determinant (Dy, etc.)
The value of the variable is the ratio of the variable
determinant to the original determinant.
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Cramers Rule Example: Solve the system using Cramers Rule.
5
6 1
7 1
8
x y
x y
+ =
+ =
40 47
6 8
2 25
D = = =
71
18 7 15
8x
D = = =
( )15
65 6 11
1yD = = =
= = =
157.5
2
xDxD
= = =
11 5.52
yDyD
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Homework College Algebra & Trigonometry
Page 862: 5-14 HW: 5-14
Page 874: 8-36 (4s), 38, 62, 66, 74
HW: 8, 16, 20, 38, 62 Classwork: Algebra & Trigonometry
Page 141: 1-8, 13 (write out augmented matrix)
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