10.2 Day 2 Systems of Linear Equations ­ Matrices 2010
December 06, 2010
10.2 Systems of Linear Equations: Matrices Day 2
Objectives:
1. Write the augmented matrix of a system of linear equations. 2. Write the system from the augmented matrix.
3. Perform row operations on a matrix.
4. Solve a system of linear equations using matrices. Nov 9­12:20 PM
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10.2 Day 2 Systems of Linear Equations ­ Matrices 2010
December 06, 2010
Warm­up:
Write the augmented matrix for each system of linear equations.
{
1.
{
5x + 7y = ­1 2.
­4x ­ 2y = 0
2y + z = 7
x + 4y ­ 8 = 0
­x ­ z ­ 9 = 0
Dec 6­12:13 PM
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10.2 Day 2 Systems of Linear Equations ­ Matrices 2010
December 06, 2010
Review: Row Operations on a Matrix Three basic row operations:
1. Interchange any two rows.
2. Replace a row by a nonzero multiple of that row.
3. Replace a row by the sum of that row and a constant nonzero multiple of some other row.
Nov 30­12:39 PM
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10.2 Day 2 Systems of Linear Equations ­ Matrices 2010
December 06, 2010
Solve a System of Linear Equations Using Matrices How do we solve a system of linear equations?
Use row operations to get to...
row
echelon
form
Example:
[ ]
[ ]
1 ­1 1 8
0 1 ­12 ­15
0 0 1 1
Reduced row echelon form:
1 0 0 4
0 1 0 ­3
0 0 1 1
Nov 30­12:40 PM
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10.2 Day 2 Systems of Linear Equations ­ Matrices 2010
December 06, 2010
Solve a System of Linear Equations Using Matrices Example #1: Given reduced row echelon form write the system of equations corresponding to the given matrix. Consistent or inconsistent? If consistent, give the solution.
[ ]
1 0 0 4
0 1 0 ­3
0 0 1 1
Answer:
Consistent
x = 4, y = ­3, z = 1
Nov 30­12:40 PM
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10.2 Day 2 Systems of Linear Equations ­ Matrices 2010
December 06, 2010
Solve a System of Linear Equations Using Matrices Example #2: Given reduced row echelon form write the system of equations corresponding to the given matrix. Consistent or inconsistent? If consistent, give the solution.
[ ]
1 0 0 ­5
0 1 0 4
0 0 0 3
Answer:
Inconsistent
Nov 30­12:40 PM
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10.2 Day 2 Systems of Linear Equations ­ Matrices 2010
December 06, 2010
Solve a System of Linear Equations Using Matrices Example #3: Given reduced row echelon form write the system of equations corresponding to the given matrix. Consistent or inconsistent? If consistent, give the solution.
[ ]
1 0 2 1
0 1 ­5 ­3
0 0 0 0
Answer:
Consistent
x = 1 ­ 2z y = ­3 + 5z z = any real number
Nov 30­12:40 PM
7
10.2 Day 2 Systems of Linear Equations ­ Matrices 2010
December 06, 2010
Solve a System of Linear Equations Using Matrices Example #4: Given reduced row echelon form write the system of equations corresponding to the given matrix. Consistent or inconsistent? If consistent, give the solution.
Answer:
[
[
1 0 0 0 2
0 1 0 2 3
0 0 1 3 0 Consistent
x1 = 2 x2 = 3 ­ 2x4 x3 = ­3x4
x4 = any real number
Nov 30­12:40 PM
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10.2 Day 2 Systems of Linear Equations ­ Matrices 2010
December 06, 2010
Homework: page 754 (25 ­ 30, 34 ­ 36, 37 ­ 59 odd) Nov 30­12:40 PM
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