Download Section 2.1

Solving System of Linear Equations
1.
2.
3.
4.
5.
6.
7.
8.
Diagonal Form of a System of Equations
Elementary Row Operations
Elementary Row Operation 1
Elementary Row Operation 2
Elementary Row Operation 3
Gaussian Elimination Method
Matrix Form of an Equation
Using Spreadsheet to Solve System
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

A system of equations is in diagonal form if each
variable only appears in one equation and only
one variable appears in an equation.
For example:
x




y
 50
 125
z  50
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

Elementary row operations are operations on the
equations (rows) of a system that alters the
system but does not change the solutions.
Elementary row operations are often used to
transform a system of equations into a diagonal
system whose solution is simple to determine.
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
Elementary Row Operation 1 Rearrange the
equations in any order.
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
Rearrange the equations of the system


3 x 
6 x


y  z  0
y  z  6
 z  3
so that all the equations containing x are on
top.
6 x

[1]  [3]

 3 x 


 z  3
y  z  6
y  z  0
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
Elementary Row Operation 2 Multiply an
equation by a nonzero number.
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
Multiply the first row of the system
6 x

3 x 



 z  3
y  z  6
y  z  0
so that the coefficient of x is 1.
x

1
[1]


6
 3x 



y 
y 
 16  z

z
z


1
2
6
0
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
Elementary Row Operation 3 Change an
equation by adding to it a multiple of another
equation.
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
Add a multiple of one row to another to
change  x
 1 z  1


3x 



 6
y 
y 
z
z


 6
 32 z
2
6
0
so that only the xfirst equation
 1 has
z an 1x term.


[2]  3[1]
 




y 
y 
z
2
 9

2
0
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
1.
2.
3.
Gaussian Elimination Method transforms a
system of linear equations into diagonal form
by repeated applications of the three
elementary row operations.
Rearrange the equations in any order.
Multiply an equation by a nonzero number.
Change an equation by adding to it a multiple
of another equation.
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
Continue Gaussian Elimination to transform
into diagonal
 x form  1 z  1






x


 1[2]
 




y 
y 

y 
y 
 6
 32 z
z
 16  z
 32 z
z
2
 9

2
0.

1
2
 9
2

0
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x







y 
y 
 16  z
 32 z
z
x
2


[3]   1[2]
 9 
 
2


0



x

2 [3]


5






y 
 16  z
 32 z
z
 16  z
 32 z
 52 z

1
y 


1
2
 9
2
 9
2
1
2
 9
2
 9
5
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x







y 
 16  z
 32 z
z

1
2
 9
2
 9
5



[1] 1 [3]
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x

[2]  3 [3]

2





x







y

 16  z
y
z

1
2
 9
5
 9
2
4
5
 9
5
z  9
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The solution is (x,y,z) = (4/5,-9/5,9/5).
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



It is often easier to do row operations if the
coefficients and constants are set up in a table
(matrix).
Each row represents an equation.
Each column represents a variable’s coefficients
except the last which represents the constants.
Such a table is called the augmented matrix of
the system of equations.
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
Write the augmented matrix for the system


3 x 
6 x

0 1 1 0 
 3 1 1 6 


6 0 1 3
y  z  0
y  z  6
 z  3.
Note:
The vertical line separates
numbers that are on opposite sides of
the equal sign.
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
The three elementary row operations for a
system of linear equations (or a matrix) are as
follows:
Rearrange the equations (rows) in any order;
 Multiply an equation (row) by a nonzero number;
 Change an equation (row) by adding to it a multiple
of another equation (row).

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