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------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/master08.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/preview.wmf Content-Transfer-Encoding: base64 Content-Type: image/x-wmf AQAJAAAD9AAAAAUASQAAAAAABAAAAAMBCAAFAAAACwIAAAAABQAAAAwCeQChAAMAAAAeAAcAAAD8 AgAA////AAAABAAAAC0BAAAIAAAA+gIFAAAAAAD///8ABAAAAC0BAQAOAAAAJAMFAP///////3gA oAB4AKAA////////CAAAAPoCAAAAAAAAAAAAAAQAAAAtAQIABAAAAC0BAAAEAAAAJwH//xwAAAD7 Avn/AAAAAAAAkAEAAAAAAEAAAEFyaWFsAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABAAAAC0B AwAEAAAALgEYAAQAAAACAQEABQAAAAkCAAAAAkkAAAAyCggACgAsAAAAU3BlY2lhbCBDYXNlcyBv ZiBTeXN0ZW1zIG9mIExpbmVhciBFcXVhdGlvbnMFAAQABAAEAAIAAwACAAIABQAEAAQAAwAEAAIA BAACAAIABQADAAQAAgAEAAYAAwACAAQAAgACAAQAAgAEAAQABAACAAIABQAEAAQABAABAAIABAAE AAMABAAAAC4BAAAcAAAA+wIQAAcAAAAAALwCAAAAAAECAiJTeXN0ZW0AAAAAAAAAAAAAGAAAAAEA AAAA8xsA5AQAAAQAAAAtAQQABAAAAPABAwADAAAAAAA= ------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0108.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Special Cases of Systems of Linear Equations
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0241.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
There are= three possible outcomes for solution of
a given s= ystem of linear equations.
Special Cases of Systems of Linear Equations
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0275.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
There are= three possible outcomes for solution of
a given s= ystem of linear equations.
I. There = is exactly one set of solution.
II. There= is no solution.
III. Ther= e are infinitely many solutions.
Special Cases of Systems of Linear Equations
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0276.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
There are= three possible outcomes for solution of
a given s= ystem of linear equations.
I. There = is exactly one set of solution.
II. There= is no solution.
III. Ther= e are infinitely many solutions.
Special Cases of Systems of Linear Equations
Below are examples of case II and III.
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0277.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
There are= three possible outcomes for solution of
a given s= ystem of linear equations.
I. There = is exactly one set of solution.
II. There= is no solution.
III. Ther= e are infinitely many solutions.
Special Cases of Systems of Linear Equations
Incons= istent (Contradictory) Systems
Below are examples of case II and III.
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0278.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
There are= three possible outcomes for solution of
a given s= ystem of linear equations.
I. There = is exactly one set of solution.
II. There= is no solution.
III. Ther= e are infinitely many solutions.
Special Cases of Systems of Linear Equations
Incons= istent (Contradictory) Systems
{
x + y =3D 2 =
x + y = =3D 3    
(E1)      
(E2)      
Example A:
Below are examples of case II and III.
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0279.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
There are= three possible outcomes for solution of
a given s= ystem of linear equations.
I. There = is exactly one set of solution.
II. There= is no solution.
III. Ther= e are infinitely many solutions.
Special Cases of Systems of Linear Equations
Incons= istent (Contradictory) Systems
{
x + y =3D 2 =
x + y = =3D 3    
Impossible! There is no solution because = the equations a= re contradictory.
(E1)      
(E2)      
Example A:
Below are examples of case II and III.
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0280.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
There are= three possible outcomes for solution of
a given s= ystem of linear equations.
I. There = is exactly one set of solution.
II. There= is no solution.
III. Ther= e are infinitely many solutions.
Special Cases of Systems of Linear Equations
Incons= istent (Contradictory) Systems
{
x + y =3D 2 =
x + y = =3D 3    
E1 – E2:    x + y =3D 2
        = ;       )  x + y = =3D 3    
0 =3D -1       
Impossible! There is no solution because = the equations a= re contradictory.
(E1)      
(E2)      
Example A:
Below are examples of case II and III.
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0281.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
There are= three possible outcomes for solution of
a given s= ystem of linear equations.
I. There = is exactly one set of solution.
II. There= is no solution.
III. Ther= e are infinitely many solutions.
Special Cases of Systems of Linear Equations
Incons= istent (Contradictory) Systems
{
x + y =3D 2 =
x + y = =3D 3    
E1 – E2:    x + y =3D 2
        = ;       )  x + y = =3D 3    
0 =3D -1       
Impossible! There= is no solution because the equations a= re contradictory.  These syste= ms are called
inconsistent or contradictory =
 systems      
(E1)      
(E2)      
Example A:
Below are examples of case II and III.
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0262.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Special Cases of Systems of Linear Equations
When solving system using matrix notation, we may <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>conclude the system is inconsistent when a row shows up that translates into 0= =3D nonzero number.
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0282.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Special Cases of Systems of Linear Equations
When solving system using matrix notation, we may <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>conclude the system is inconsistent when a row shows up that translates into 0= =3D nonzero number.
For example, the system corresp= onds to        = ;            &n= bsp;
is inconsistent.
*   *   *   *
-2=
         *   *
0
0
0
0
0
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0283.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Depend= ent Systems
Special Cases of Systems of Linear Equations
When solving system using matrix notation, we may <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>conclude the system is inconsistent when a row shows up that translates into 0= =3D nonzero number.
For example, the system corresp= onds to        = ;            &n= bsp;
is inconsistent.
*   *   *   *
-2=
         *   *
0
0
0
0
0
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0284.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Depend= ent Systems
Special Cases of Systems of Linear Equations
Example B:
{
x + y =3D 2 =
2x + 2y = =3D 4    
(E1)      
(E2)      
When solving system using matrix notation, we may <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>conclude the system is inconsistent when a row shows up that translates into 0= =3D nonzero number.
For example, the system corresp= onds to        = ;            &n= bsp;
is inconsistent.
*   *   *   *
-2=
         *   *
0
0
0
0
0
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0285.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Depend= ent Systems
The equatio= ns are the same. = ;
Special Cases of Systems of Linear Equations
Example B:
{
x + y =3D 2 =
2x + 2y = =3D 4    
(E1)      
(E2)      
When solving system using matrix notation, we may <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>conclude the system is inconsistent when a row shows up that translates into 0= =3D nonzero number.
For example, the system corresp= onds to        = ;            &n= bsp;
is inconsistent.
*   *   *   *
-2=
         *   *
0
0
0
0
0
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0286.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Depend= ent Systems
The equations are= the same. This system has infinitely many soluti= ons such as
(3, 0), (2, 1),and (1, 2). 
Special Cases of Systems of Linear Equations
Example B:
{
x + y =3D 2 =
2x + 2y = =3D 4    
(E1)      
(E2)      
When solving system using matrix notation, we may <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>conclude the system is inconsistent when a row shows up that translates into 0= =3D nonzero number.
For example, the system corresp= onds to        = ;            &n= bsp;
is inconsistent.
*   *   *   *
-2=
         *   *
0
0
0
0
0
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0287.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Depend= ent Systems
The equations are= the same. This system has infinitely many soluti= ons such as
(3, 0), (2, 1),and (1, 2). 
Such systems are called dependent systems.
Special Cases of Systems of Linear Equations
Example B:
{
x + y =3D 2 =
2x + 2y = =3D 4    
(E1)      
(E2)      
When solving system using matrix notation, we may <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>conclude the system is inconsistent when a row shows up that translates into 0= =3D nonzero number.
For example, the system corresp= onds to        = ;            &n= bsp;
is inconsistent.
*   *   *   *
-2=
         *   *
0
0
0
0
0
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0288.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Depend= ent Systems
The equations are= the same. This system has infinitely many soluti= ons such as
(3, 0), (2, 1),and (1, 2). 
Such systems are called dependent systems.
Special Cases of Systems of Linear Equations
Example B:
{
x + y =3D 2 =
2x + 2y = =3D 4    
2*E1 – E2:  
        2x + 2y =3D 4
      ) 2x + 2y =3D 4     <= /span>
0 =3D 0       
(E1)      
(E2)      
When solving system using matrix notation, we may <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>conclude the system is inconsistent when a row shows up that translates into 0= =3D nonzero number.
For example, the system corresp= onds to        = ;            &n= bsp;
is inconsistent.
*   *   *   *
-2=
         *   *
0
0
0
0
0
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0261.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Special Cases of Systems of Linear Equations
A system is dependent if the it is not inconsistent and <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>the system has more varia= bles than equations.
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0289.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Special Cases of Systems of Linear Equations
A system is dependent if the it is not inconsistent and <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>the system has more varia= bles than equations.
This means that there is = not sufficient condition to determine one solution. =
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0290.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Special Cases of Systems of Linear Equations
A system is dependent if the it is not inconsistent and <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>the system has more varia= bles than equations.
This means that there is not sufficient condition to determine one solution.
In the case of linear systems, = this means that there'll be infinite many solutions .
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0291.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Special Cases of Systems of Linear Equations
A system is dependent if the it is not inconsistent and <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>the system has more varia= bles than equations.
This means that there is not sufficient condition to determine one solution.
In the case of linear systems, = this means that there'll be infinite many solutions . These solutions may be <= span style=3D'position:absolute;top:46.25%;left:4.3%;width:108.05%;height:6.0%'= >given by expressing some = of the variable in terms of the other variables.
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0292.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Special Cases of Systems of Linear Equations
A system is dependent if the it is not inconsistent and <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>the system has more varia= bles than equations.
This means that there is not sufficient condition to determine one solution.
In the case of linear systems, = this means that there'll be infinite many solutions . These solutions may be <= span style=3D'position:absolute;top:46.25%;left:4.3%;width:108.05%;height:6.0%'= >given by expressing some = of the variable in terms of the other variables.
Example C:  The system
in example B is dependent.
{
x + y =3D 2 =
2x + 2y = =3D 4    
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0293.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Special Cases of Systems of Linear Equations
A system is dependent if the it is not inconsistent and <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>the system has more varia= bles than equations.
This means that there is not sufficient condition to determine one solution.
In the case of linear systems, = this means that there'll be infinite many solutions . These solutions may be <= span style=3D'position:absolute;top:46.25%;left:4.3%;width:108.05%;height:6.0%'= >given by expressing some = of the variable in terms of the other variables.
Example C:  The system
in example B is dependent.
{
x + y =3D 2 =
2x + 2y = =3D 4    
There is re= ally only one equation x + y =3D 2.
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0294.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Special Cases of Systems of Linear Equations
A system is dependent if the it is not inconsistent and <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>the system has more varia= bles than equations.
This means that there is not sufficient condition to determine one solution.
In the case of linear systems, = this means that there'll be infinite many solutions . These solutions may be <= span style=3D'position:absolute;top:46.25%;left:4.3%;width:108.05%;height:6.0%'= >given by expressing some = of the variable in terms of the other variables.
Example C:  The system
in example B is dependent.
{
x + y =3D 2 =
2x + 2y = =3D 4    
There is really o= nly one equation x + y =3D 2.
We may express, s= ay y, in term of x as y =3D 2 – x.
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0295.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Special Cases of Systems of Linear Equations
A system is dependent if the it is not inconsistent and <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>the system has more varia= bles than equations.
This means that there is not sufficient condition to determine one solution.
In the case of linear systems, = this means that there'll be infinite many solutions . These solutions may be <= span style=3D'position:absolute;top:46.25%;left:4.3%;width:108.05%;height:6.0%'= >given by expressing some = of the variable in terms of the other variables.
Example C:  The system
in example B is dependent.
{
x + y =3D 2 =
2x + 2y = =3D 4    
There is really o= nly one equation x + y =3D 2.
We may express, s= ay y, in term of x as y =3D 2 – x.
So the solutions = are (x, 2 – x) where x is any number.
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0296.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Special Cases of Systems of Linear Equations
A system is dependent if the it is not inconsistent and <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;mso-fareast-language:ZH-CN'>the system has more varia= bles than equations.
This means that there is not sufficient condition to determine one solution.
In the case of linear systems, = this means that there'll be infinite many solutions . These solutions may be <= span style=3D'position:absolute;top:46.25%;left:4.3%;width:108.05%;height:6.0%'= >given by expressing some = of the variable in terms of the other variables.
Example C:  The system
in example B is dependent.
{
x + y =3D 2 =
2x + 2y = =3D 4    
There is really o= nly one equation x + y =3D 2.
We may express, s= ay y, in term of x as y =3D 2 – x.
So the solutions = are (x, 2 – x) where x is any number.
For instance, if = we set x =3D 3, then (3, -1) is a solution.
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Special Cases of Systems of Linear Equations
To write the solutions of dependent systems, reduce = the matrices into the row reduced echelon form (rref).
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0263.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Special Cases of Systems of Linear Equations
To write the solutions of dependent systems, reduce = the matrices into the row reduced echelon form (rref).
The rref-form of a matrix is the further row reduction of the upper diagonal form by making
1st. t= he first nonzero entries of any row to the right   &#= 13;
      of the first= nonzero entries of the previous row.
------=_NextPart_01C8942F.2778EFB0 Content-Location: file:///C:/09764A33/4SpecialCasesSystemofLinearEquations_files/slide0297.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Special Cases of Systems of Linear Equations
To write the solutions of dependent systems, reduce = the matrices into the row reduced echelon form (rref).
The rref-form of a matrix is the further row reduction of the upper diagonal form by making
1st. t= he first nonzero entries of any row to the right   &#= 13;
      of the first= nonzero entries of the previous row.
2nd. a= ll the first nonzero entries in each rows as 1.
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Special Cases of Systems of Linear Equations
To write the solutions of dependent systems, reduce = the matrices into the row reduced echelon form (rref).
The rref-form of a matrix is the further row reduction of the upper diagonal form by making
1st. t= he first nonzero entries of any row to the right   &#= 13;
      of the first= nonzero entries of the previous row.
2nd. a= ll the first nonzero entries in each rows as 1.
3rd. 0= 's in all the positions directly over the leading 1's.
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Special Cases of Systems of Linear Equations
To write the solutions of dependent systems, reduce = the matrices into the row reduced echelon form (rref).
The rref-form of a matrix is the further row reduction of the upper diagonal form by making
1st. t= he first nonzero entries of any row to the right   &#= 13;
      of the first= nonzero entries of the previous row.
2nd. a= ll the first nonzero entries in each rows as 1.
3rd. 0= 's in all the positions directly over the leading 1's.
2      1     2     4  
Example D: Row reduce
A into rref-form where
 A =3D
0      0     2     0 
0      0    -4     4 
0      0     0     2  
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Special Cases of Systems of Linear Equations
2      1     2     4  
0      0     2     0 
0      0    -4     4 
0      0     0     2  
A is an upper diagonal matrix.
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Special Cases of Systems of Linear Equations
2      1     2     4  
0      0     2     0 
0      0    -4     4 
0      0     0     2  
A is an upper diagonal matrix. For it to be in the
rref-form, the leading nonzero number in the 3rd has
to be to the right of the leading "2" o= f the 2nd row.
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Special Cases of Systems of Linear Equations
2      1     2     4  
0      0     2     0 
0      0    -4     4 
0      0     0     2  
2*R2 add to R= 3
0         0         4         0 
A is an upper diagonal matrix. For it to be in the
rref-form, the leading nonzero number in the 3rd has
to be to the right of the leading "2" of the 2nd row.
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Special Cases of Systems of Linear Equations
2      1     2     4  
0      0     2     0 
0      0    -4     4 
0      0     0     2  
2*R2 add to R= 3
2      1     2     4  
0      0     2     0 
0      0     0     4 
0      0     0     2  
0         0         4         0 
A is an upper diagonal matrix. For it to be in the
rref-form, the leading nonzero number in the 3rd has
to be to the right of the leading "2" of the 2nd row.
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Special Cases of Systems of Linear Equations
2      1     2     4  
0      0     2     0 
0      0    -4     4 
0      0     0     2  
2*R2 add to R= 3
2      1     2     4  
0      0     2     0 
0      0     0     4 
0      0     0     2  
0         0         4         0 
We can't have the leading
"2" in the 4 row below
the leading "4" of the 3rd 
row so we (-½)*R3 add to R4.
A is an upper diagonal matrix. For it to be in the
rref-form, the leading nonzero number in the 3rd has
to be to the right of the leading "2" of the 2nd row.
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Special Cases of Systems of Linear Equations
2      1     2     4  
0      0     2     0 
0      0    -4     4 
0      0     0     2  
2*R2 add to R= 3
2      1     2     4  
0      0     2     0 
0      0     0     4 
0      0     0     2  
0         0         4         0 
We can't have the leading
"2" in the 4 row below
the leading "4" of the 3rd 
row so we (-½)*R3 add to R4.
(- ½ )*R3 add to R4
2      1     2     4  
0      0     2     0 
0      0     0     4 
0      0     0     0  
A is an upper diagonal matrix. For it to be in the
rref-form, the leading nonzero number in the 3rd has
to be to the right of the leading "2" of the 2nd row.
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Special Cases of Systems of Linear Equations
2      1     2     4  
0      0     2     0 
0      0     0     4 
0      0     0     0  
Next step is to make all the leading nonzero entries = <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:#003300;mso-fareast-language:ZH-CN'>into 1's. <= /span>
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Special Cases of Systems of Linear Equations
2      1     2     4