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2. Starti= ng from the bottom row, get the answer for one of the = variable.=
Matrix Notation<= /span>
------=_NextPart_01C8FB43.DE4A21C0 Content-Location: file:///C:/AE4BB24E/5MatrixNotation_files/slide0087.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Matrix Notation
2. Starti= ng from the bottom row, get the answer for one of the = variable.= Then go up one row, using the solution already obtained to get another answer of another variable.
Matrix Notation<= /span>
------=_NextPart_01C8FB43.DE4A21C0 Content-Location: file:///C:/AE4BB24E/5MatrixNotation_files/slide0088.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Matrix Notation
2. Starti= ng from the bottom row, get the answer for one of the = variable.= Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working u= p the rows to extract all solutions. 
Matrix Notation<= /span>
------=_NextPart_01C8FB43.DE4A21C0 Content-Location: file:///C:/AE4BB24E/5MatrixNotation_files/slide0089.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Matrix Notation
2. Starti= ng from the bottom row, get the answer for one of the = variable.= Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working u= p the rows to extract all solutions. 
Matrix Notation<= /span>
  x + 4y =3D -7
 2x – 3y =3D 8    
{
Eq. 1
Eq. 2
Example A. Solve using matrix notation.
------=_NextPart_01C8FB43.DE4A21C0 Content-Location: file:///C:/AE4BB24E/5MatrixNotation_files/slide0089_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFQAEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAS AAIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C8FB43.DE4A21C0 Content-Location: file:///C:/AE4BB24E/5MatrixNotation_files/slide0089_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFQADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQAS AAIAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C8FB43.DE4A21C0 Content-Location: file:///C:/AE4BB24E/5MatrixNotation_files/slide0090.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Matrix Notation
2. Starti= ng from the bottom row, get the answer for one of the = variable.= Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working u= p the rows to extract all solutions. 
Matrix Notation<= /span>
  x + 4y =3D -7
 2x – 3y =3D 8    
{
Eq. 1
Eq. 2
Example A. Solve using matrix notation.
Put the system into a matrix:
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2. Starti= ng from the bottom row, get the answer for one of the = variable.= Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working u= p the rows to extract all solutions. 
Matrix Notation<= /span>
   1    4    -7
   2   -3     8
  x + 4y =3D -7
 2x – 3y =3D 8    
{
Eq. 1
Eq. 2
Example A. Solve using matrix notation.
Put the system into a matrix:
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2. Starti= ng from the bottom row, get the answer for one of the = variable.= Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working u= p the rows to extract all solutions. 
Matrix Notation<= /span>
   1    4    -7
   2   -3     8
  x + 4y =3D -7
 2x – 3y =3D 8    
{
Eq. 1
Eq. 2
Example A. Solve using matrix notation.
Put the system into a matrix:
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2. Starti= ng from the bottom row, get the answer for one of the = variable.= Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working u= p the rows to extract all solutions. 
Matrix Notation<= /span>
   1    4    -7
   2   -3     8
 -2*= R1 add à R2
  x + 4y =3D -7
 2x – 3y =3D 8    
{
Eq. 1
Eq. 2
Example A. Solve using matrix notation.
Put the system into a matrix:
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2. Starti= ng from the bottom row, get the answer for one of the = variable.= Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working u= p the rows to extract all solutions. 
Matrix Notation<= /span>
   1    4    -7
   2   -3     8
 -2*= R1 add à R2
 -2<= span style=3D'mso-spacerun:yes'>     -8      14=
  x + 4y =3D -7
 2x – 3y =3D 8    
{
Eq. 1
Eq. 2
Example A. Solve using matrix notation.
Put the system into a matrix:
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2. Starti= ng from the bottom row, get the answer for one of the = variable.= Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working u= p the rows to extract all solutions. 
Matrix Notation<= /span>
   1    4    -7
   2   -3     8
 -2*= R1 add à R2
<= span style=3D'font-size:133%;color:#003300'>1   4   -7
0  -11  22
 =
 -2<= span style=3D'mso-spacerun:yes'>     -8      14=
  x + 4y =3D -7
 2x – 3y =3D 8    
{
Eq. 1
Eq. 2
Example A. Solve using matrix notation.
Put the system into a matrix:
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2. Starti= ng from the bottom row, get the answer for one of the = variable.= Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working u= p the rows to extract all solutions. 
Matrix Notation<= /span>
   1    4    -7
   2   -3     8
 -2*= R1 add à R2
<= span style=3D'font-size:133%;color:#003300'>1   4   -7
0  -11  22
 =
 -2<= span style=3D'mso-spacerun:yes'>     -8      14=
  x + 4y =3D -7
 2x – 3y =3D 8    
{
Eq. 1
Eq. 2
Example A. Solve using matrix notation.
Put the system into a matrix:
From the bottom R2:  -11y =3D 22 à y =3D -2
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2. Starti= ng from the bottom row, get the answer for one of the = variable.= Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working u= p the rows to extract all solutions. 
Matrix Notation<= /span>
   1    4    -7
   2   -3     8
 -2*= R1 add à R2
<= span style=3D'font-size:133%;color:#003300'>1   4   -7
0  -11  22
 =
 -2<= span style=3D'mso-spacerun:yes'>     -8      14=
  x + 4y =3D -7
 2x – 3y =3D 8    
{
Eq. 1
Eq. 2
Example A. Solve using matrix notation.
Put the system into a matrix:
From the bottom R2:  -11y =3D 22 à y =3D -2
Go up one row to R1 and set y =3D -2:
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2. Starti= ng from the bottom row, get the answer for one of the = variable.= Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working u= p the rows to extract all solutions. 
Matrix Notation<= /span>
   1    4    -7
   2   -3     8
 -2*= R1 add à R2
<= span style=3D'font-size:133%;color:#003300'>1   4   -7
0  -11  22
 =
 -2<= span style=3D'mso-spacerun:yes'>     -8      14=
  x + 4y =3D -7
 2x – 3y =3D 8    
{
Eq. 1
Eq. 2
Example A. Solve using matrix notation.
Put the system into a matrix:
From the bottom R2:  -11y =3D 22 à y =3D -2
Go up one row to R1 and set y =3D -2:
x + 4y =3D -7
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2. Starti= ng from the bottom row, get the answer for one of the = variable.= Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working u= p the rows to extract all solutions. 
Matrix Notation<= /span>
   1    4    -7
   2   -3     8
 -2*= R1 add à R2
<= span style=3D'font-size:133%;color:#003300'>1   4   -7
0  -11  22
 =
 -2<= span style=3D'mso-spacerun:yes'>     -8      14=
  x + 4y =3D -7
 2x – 3y =3D 8    
{
Eq. 1
Eq. 2
Example A. Solve using matrix notation.
Put the system into a matrix:
From the bottom R2:  -11y =3D 22 à y =3D -2
Go up one row to R1 and set y =3D -2:
x + 4y =3D -7
x + 4(-2) =3D -7
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2. Starti= ng from the bottom row, get the answer for one of the = variable.= Then go up one row, using the solution already obtained to get another answer of another variable. Repeat the process, working u= p the rows to extract all solutions. 
Matrix Notation<= /span>
   1    4    -7
   2   -3     8
 -2*= R1 add à R2
<= span style=3D'font-size:133%;color:#003300'>1   4   -7
0  -11  22
 =
 -2<= span style=3D'mso-spacerun:yes'>     -8      14=
  x + 4y =3D -7
 2x – 3y =3D 8    
{
Eq. 1
Eq. 2
Example A. Solve using matrix notation.
Put the system into a matrix:
From the bottom R2:  -11y =3D 22 à y =3D -2
Go up one row to R1 and set y =3D -2:
x + 4y =3D -7
x + 4(-2) =3D -7
x =3D 1        = ;             Solution : (1, -2)
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Example B. Solve using matrix notation.
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
Matrix Notation<= /span>
------=_NextPart_01C8FB43.DE4A21C0 Content-Location: file:///C:/AE4BB24E/5MatrixNotation_files/slide0102.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Matrix Notation
Example B. Solve using matrix notation.
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
Put the system into a matrix:
Matrix Notation<= /span>
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Example B. Solve using matrix notation.
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
2     3      3     13
Put the system into a matrix:
1     2      2      8<= /div>
3     2      3     13
Matrix Notation<= /span>
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Example B. Solve using matrix notation.
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
2     3      3     13
    R1       R2<= /span>
Put the system into a matrix:
1     2      2      8<= /div>
3     2      3     13
Matrix Notation<= /span>
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Example B. Solve using matrix notation.
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
2     3      3     13
    R1       R2<= /span>
Put the system into a matrix:
1     2      2      8<= /div>
3     2      3     13
2     3      3     13
1     2      2      8<= /div>
3     2      3     13
Matrix Notation<= /span>
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Example B. Solve using matrix notation.
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
2     3      3     13
    R1       R2<= /span>
Put the system into a matrix:
1     2      2      8<= /div>
3     2      3     13
2     3      3     13
1     2      2      8<= /div>
3     2      3     13
-2R1 add  R2
Matrix Notation<= /span>
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Example B. Solve using matrix notation.
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
2     3      3     13
    R1       R2<= /span>
-2     -4      -4     -16
Put the system into a matrix:
1     2      2      8<= /div>
3     2      3     13
2     3      3     13
1     2      2      8<= /div>
3     2      3     13
-2R1 add  R2
Matrix Notation<= /span>
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Example B. Solve using matrix notation.
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
2     3      3     13
    R1       R2<= /span>
-2     -4      -4     -16
Put the system into a matrix:
1     2      2      8<= /div>
3     2      3     13
2     3      3     13
1     2      2      8<= /div>
3     2      3     13
0    -1     -1     -3
1     2      2      8<= /div>
3     2      3     13
-2R1 add  R2
Matrix Notation<= /span>
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Example B. Solve using matrix notation.
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
2     3      3     13
    R1       R2<= /span>
-2     -4      -4     -16
Put the system into a matrix:
1     2      2      8<= /div>
3     2      3     13
2     3      3     13
1     2      2      8<= /div>
3     2      3     13
0    -1     -1     -3
1     2      2      8<= /div>
3     2      3     13
-3* R1 add R3
-2R1 add  R2
Matrix Notation<= /span>
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image/gif R0lGODlhXAALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAwBZ AAUAgAAAAAAAAAIghI+py+0PUwix2ovhnGb7D4biSJbmiabqyorZC8cLdxQAOx== ------=_NextPart_01C8FB43.DE4A21C0 Content-Location: file:///C:/AE4BB24E/5MatrixNotation_files/slide0109.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Matrix Notation
Example B. Solve using matrix notation.
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
2     3      3     13
    R1       R2<= /span>
-2     -4      -4     -16
Put the system into a matrix:
1     2      2      8<= /div>
3     2      3     13
2     3      3     13
1     2      2      8<= /div>
3     2      3     13