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Systems of Linear Equations
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Systems of Linear Equations
We = need one piece of information to solve for one <= 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'>unknown quantity. =
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Systems of Linear Equations
We = need one piece of information to solve for one = unknown quantity. If 2 hamburgers cost 4 dollars and x is the cost of a hamburger, t= hen 2x =3D 4 or x =3D $2.
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Systems of Linear Equations
We need one piece of informatio= n to solve for one unknown quantity. If 2 hamburgers cost 4 dollars and = x is the cost of a hambur= ger, then 2x =3D 4 or x =3D $2.
If there are two unknowns x and= y, we need two pieces of information to solve them.
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Systems of Linear Equations
We need one piece of informatio= n to solve for one unknown quantity. If 2 hamburgers cost 4 dollars and = x is the cost of a hambur= ger, then 2x =3D 4 or x =3D $2.
If there are two unknowns x and= y, we need two pieces of information to solve them.
Example A: <= /span>
Let x =3D cost of= a hamburger,  y =3D cost of a f= ry. If two hamburgers and a fry cost $7, and one hamburge= r and one fry cost $5. what is the price of each?
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Systems of Linear Equations
We need one piece of informatio= n to solve for one unknown quantity. If 2 hamburgers cost 4 dollars and = x is the cost of a hambur= ger, then 2x =3D 4 or x =3D $2.
If there are two unknowns x and= y, we need two pieces of information to solve them.
Example A: <= /span>
Let x =3D cost of= a hamburger,  y =3D cost of a f= ry. If two hamburgers and a fry cost $7, and one hamburge= r and one fry cost $5. what is the price of each?
We translate these information in to two equations:
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Systems of Linear Equations
We need one piece of informatio= n to solve for one unknown quantity. If 2 hamburgers cost 4 dollars and = x is the cost of a hambur= ger, then 2x =3D 4 or x =3D $2.
If there are two unknowns x and= y, we need two pieces of information to solve them.
Example A: <= /span>
Let x =3D cost of= a hamburger,  y =3D cost of a f= ry. If two hamburgers and a fry cost $7, and one hamburge= r and one fry cost $5. what is the price of each?
 2x + y =3D 7
     
We translate these information in to two equations:
Eq. 1
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Systems of Linear Equations
We need one piece of informatio= n to solve for one unknown quantity. If 2 hamburgers cost 4 dollars and = x is the cost of a hambur= ger, then 2x =3D 4 or x =3D $2.
If there are two unknowns x and= y, we need two pieces of information to solve them.
Example A: <= /span>
Let x =3D cost of= a hamburger,  y =3D cost of a f= ry. If two hamburgers and a fry cost $7, and one hamburge= r and one fry cost $5. what is the price of each?
 2x + y =3D 7
   x + y = =3D 5    
{
Eq. 1
Eq. 2
We translate these information in to two equations:
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Systems of Linear Equations
We need one piece of informatio= n to solve for one unknown quantity. If 2 hamburgers cost 4 dollars and = x is the cost of a hambur= ger, then 2x =3D 4 or x =3D $2.
If there are two unknowns x and= y, we need two pieces of information to solve them.
Example A: <= /span>
Let x =3D cost of= a hamburger,  y =3D cost of a f= ry. If two hamburgers and a fry cost $7, and one hamburge= r and one fry cost $5. what is the price of each?
 2x + y =3D 7
   x + y = =3D 5    
{
A group of equations such as this is called a system = <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:red;mso-fareast-language:ZH-CN'>of equations= .
Eq. 1
Eq. 2
We translate these information in to two equations:
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Systems of Linear Equations
We are to find values for x and for y that will satisfy all = = equations in the system.
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0182.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Systems of Linear Equations
We are to find values for x and for y that will satisfy all = = equations in the system. Such a set of values is called a <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:red;mso-fareast-language:ZH-CN'>solution= for the system.
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0183.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Systems of Linear Equations
We are to find values for x and for y that will satisfy all = = equations in the system. Such a set of values is called a <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:red;mso-fareast-language:ZH-CN'>solution= for the system. To solve the system, we note that the difference in the two orders is one hamburger and the difference in price is $2.
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0184.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Systems of Linear Equations
We are to find values for x and for y that will satisfy all = = equations in the system. Such a set of values is called a <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:red;mso-fareast-language:ZH-CN'>solution= for the system. To solve the system, we note that the difference in the two orders is one hamburger and the difference in price is $2. So the price of the hamburger is $2.
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0185.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Systems of Linear Equations
We are to find va= lues for x and for y that will satisfy all equations in the system. Such a set of values is called a <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:red;mso-fareast-language:ZH-CN'>solution= for the system. To solve the system, we note that the difference in the two orders is one hamburger and the difference in price is $2. So the price of the hamburger is $2. In algebra, we subtract
the equations in = the system: 
 2x + y =3D 7
   x + y = =3D 5    
)
Eq. 1
Eq. 2
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0186.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Systems of Linear Equations
We are to find va= lues for x and for y that will satisfy all equations in the system. Such a set of values is called a <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:red;mso-fareast-language:ZH-CN'>solution= for the system. To solve the system, we note that the difference in the two orders is one hamburger and the difference in price is $2. So the price of the hamburger is $2. In algebra, we subtract
the equations in = the system: 
 2x + y =3D 7
   x + y = =3D 5    
)
x       =3D= 2    
Eq. 1
Eq. 2
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0187.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Systems of Linear Equations
We are to find va= lues for x and for y that will satisfy all equations in the system. Such a set of values is called a <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:red;mso-fareast-language:ZH-CN'>solution= for the system. To solve the system, we note that the difference in the two orders is one hamburger and the difference in price is $2. So the price of the hamburger is $2. In algebra, we subtract
the equations in = the system: 
 2x + y =3D 7
   x + y = =3D 5    
)
x       =3D= 2    
Eq. 1
Eq. 2
Put x =3D 2= back into Eq. 2, we get y =3D 3.  &n= bsp;
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0188.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Systems of Linear Equations
We are to find va= lues for x and for y that will satisfy all equations in the system. Such a set of values is called a <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:red;mso-fareast-language:ZH-CN'>solution= for the system. To solve the system, we note that the difference in the two orders is one hamburger and the difference in price is $2. So the price of the hamburger is $2. In algebra, we subtract
the equations in = the system: 
 2x + y =3D 7
   x + y = =3D 5    
)
x       =3D= 2    
Eq. 1
Eq. 2
Put x =3D 2= back into Eq. 2, we get y =3D 3.  &n= bsp;
Hence a hamburger is $2 and a fry is $3.
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0189.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Systems of Linear Equations
We are to find va= lues for x and for y that will satisfy all equations in the system. Such a set of values is called a <= span lang=3DEN-US style=3D'mso-fareast-font-family:\5B8B\4F53;mso-hansi-font-fa= mily: Arial;font-size:156%;color:red;mso-fareast-language:ZH-CN'>solution= for the system. To solve the system, we note that the difference in the two orders is one hamburger and the difference in price is $2. So the price of the hamburger is $2. In algebra, we subtract
the equations in = the system: 
 2x + y =3D 7
   x + y = =3D 5    
)
x       =3D= 2    
Eq. 1
Eq. 2
Put x =3D 2= back into Eq. 2, we get y =3D 3.  &n= bsp;
Hence a hamburger is $2 and a fry is $3.
Thi= s is called the elimination method - we eliminate  variables by adding or subtracting two equations.  <= /span>
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Elimination method reduces the number of variables in the system one at a time.
Systems of Linear Equations
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Elimination method reduces the number of variables in the system one at a time. The method = changes a given system in= to a smaller system.
Systems of Linear Equations
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0191.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Elimination method reduces the number of variables in the system one at a time. The method changes a given system in= to a smaller system. We do this successively until the solution is clear.
Systems of Linear Equations
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0192.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
Elimination method reduces the number of variables in the system one at a time. The method changes a given system in= to a smaller system. We do this successively until the solution is clear.
Systems of Linear Equations
Example B:
Let x =3D cost of= a hamburger,  y =3D cost of a f= ry,
z =3D cost of a s= oda.
2 hamburgers, 3 f= ry and 3 soda cost $13.
1 hamburger, 2 fry and 2 soda cost $8.
3 hamburgers, 2 fries, 3 soda cost $13.  = ;
Find x, y, and z.=
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Elimination method reduces the number of variables in the system one at a time. The method changes a given system in= to a smaller system. We do this successively until the solution is clear.
Systems of Linear Equations
Example B:
Let x =3D cost of= a hamburger,  y =3D cost of a f= ry,
z =3D cost of a s= oda.
2 hamburgers, 3 f= ry and 3 soda cost $13.
1 hamburger, 2 fry and 2 soda cost $8.
3 hamburgers, 2 fries, 3 soda cost $13.  = ;
Find x, y, and z.=
We translate the information into a system of three <= span style=3D'position:absolute;top:89.0%;left:4.3%;width:88.95%;height:6.0%'><= 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'>equations w= ith three unknowns
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 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
Systems of Linear Equations
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 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
Select x to eliminate since there is 1x in E2.
Systems of Linear Equations
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 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
Select x to eliminate since there is 1x in E2.
-2*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 2 + E1:
Systems of Linear Equations
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0196.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
Select x to eliminate since there is 1x in E2.
-2*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 2 + E1:
      -2= x – 4y – 4z =3D -16 
+)    2x + 3y + 3z =3D 13 <= /span>
Systems of Linear Equations
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0197.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
Select x to eliminate since there is 1x in E2.
-2*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 2 + E1:
      -2= x – 4y – 4z =3D -16 
+)    2x + 3y + 3z =3D 13 <= /span>
        = ;  0 –  y  –  z  =3D - 3
Systems of Linear Equations
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0198.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
Select x to eliminate since there is 1x in E2.
-2*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 2 + E1:
      -2= x – 4y – 4z =3D -16 
+)    2x + 3y + 3z =3D 13 <= /span>
        = ;  0 –  y  –  z  =3D - 3
-3*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 2 + E3:
Systems of Linear Equations
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0199.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
Select x to eliminate since there is 1x in E2.
-2*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 2 + E1:
      -2= x – 4y – 4z =3D -16 
+)    2x + 3y + 3z =3D 13 <= /span>
        = ;  0 –  y  –  z  =3D - 3
-3*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 2 + E3:
      -3= x – 6y – 6z =3D -24 
+)    3x + 2y + 3z =3D 13 <= /span>
Systems of Linear Equations
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0200.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
Select x to eliminate since there is 1x in E2.
-2*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 2 + E1:
      -2= x – 4y – 4z =3D -16 
+)    2x + 3y + 3z =3D 13 <= /span>
        = ;  0 –  y  –  z  =3D - 3
-3*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 2 + E3:
      -3= x – 6y – 6z =3D -24 
+)    3x + 2y + 3z =3D 13 <= /span>
        = ;  0  – 4y  – 3z =3D -11
Systems of Linear Equations
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 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
Select x to eliminate since there is 1x in E2.
-2*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 2 + E1:
      -2= x – 4y – 4z =3D -16 
+)    2x + 3y + 3z =3D 13 <= /span>
        = ;  0 –  y  –  z  =3D - 3
-3*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 2 + E3:
      -3= x – 6y – 6z =3D -24 
+)    3x + 2y + 3z =3D 13 <= /span>
        = ;  0  – 4y  – 3z =3D -11
Group these into a system of two equations with = two unknown= s.
Systems of Linear Equations
------=_NextPart_01C890F6.C6135750 Content-Location: file:///C:/2F561913/3SystemofLinearEquationsandMatrices_files/slide0202.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" System of Linear Equations
 2x + 3y + 3z =3D 13
   x + 2y + 2z =3D 8 
 3x + 2y + 3z =3D 13  
{
E 1
E 2
E 3
Select x to eliminate since there is 1x in E2.
-2*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 2 + E1:
      -2= x – 4y – 4z =3D -16 
+)    2x + 3y + 3z =3D 13 <= /span>
        = ;  0 –  y  –  z  =3D - 3
-3*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 2 + E3:
      -3= x – 6y – 6z =3D -24 
+)    3x + 2y + 3z =3D 13 <= /span>
        = ;  0  – 4y  – 3z =3D -11
Group these into a system of two equations with = two unknown= s.
Systems of Linear Equations
Hence we reduced the system to a simpler one. 
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–  y  –  z  =3D - 3    &nb= sp;
– 4y  – 3z= =3D -11
{
Systems of Linear Equations
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–  y  –  z  =3D - 3    &nb= sp;
– 4y  – 3z= =3D -11
{
4y  + 3z =3D 11
{
   y  + z  =3D  3
Eq.4
Eq. 5
Systems of Linear Equations
 *(-1) 
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–  y  –  z  =3D - 3    &nb= sp;
– 4y  – 3z= =3D -11
{
4y  + 3z =3D 11
{
   y  + z  =3D  3
Eq.4
Eq. 5
Systems of Linear Equations
 *(-1) 
Let's elimi= nate y.
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–  y  –  z  =3D - 3    &nb= sp;
– 4y  – 3z= =3D -11
{
4y  + 3z =3D 11
{
   y  + z  =3D  3
Eq.4
Eq. 5
-3*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 4 + E5:
Systems of Linear Equations
 *(-1) 
Let's elimi= nate y.
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–  y  –  z  =3D - 3    &nb= sp;
– 4y  – 3z= =3D -11
{
4y  + 3z =3D 11
{
   y  + z  =3D  3
Eq.4
Eq. 5
-3*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 4 + E5:
+) 4y  + 3z =3D 11
   -3y – 3z  =3D -9
Systems of Linear Equations
 *(-1) 
Let's elimi= nate y.
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–  y  –  z  =3D - 3    &nb= sp;
– 4y  – 3z= =3D -11
{
4y  + 3z =3D 11
{
   y  + z  =3D  3
Eq.4
Eq. 5
-3*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 4 + E5:
+) 4y  + 3z =3D 11
   -3y – 3z  =3D -9
    y   +  0  =3D  2
        = ;      y =3D  2
Systems of Linear Equations
 *(-1) 
Let's elimi= nate y.
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–  y  –  z  =3D - 3    &nb= sp;
– 4y  – 3z= =3D -11
{
4y  + 3z =3D 11
{
   y  + z  =3D  3
Eq.4
Eq. 5
-3*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 4 + E5:
+) 4y  + 3z =3D 11
   -3y – 3z  =3D -9
    y   +  0  =3D  2
        = ;      y =3D  2
To get z, s= et 2 for y in E4:
Systems of Linear Equations
 *(-1) 
Let's elimi= nate y.
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–  y  –  z  =3D - 3    &nb= sp;
– 4y  – 3z= =3D -11
{
4y  + 3z =3D 11
{
   y  + z  =3D  3
Eq.4
Eq. 5
-3*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 4 + E5:
+) 4y  + 3z =3D 11
   -3y – 3z  =3D -9
    y   +  0  =3D  2
        = ;      y =3D  2
To get z, s= et 2 for y in E4:2 + z =3D 3  à z =3D 1    
Systems of Linear Equations
 *(-1) 
Let's elimi= nate y.
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–  y  –  z  =3D - 3    &nb= sp;
– 4y  – 3z= =3D -11
{
4y  + 3z =3D 11
{
   y  + z  =3D  3
Eq.4
Eq. 5
-3*<= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>E 4 + E5:
+) 4y  + 3z =3D 11
   -3y – 3z  =3D -9
    y   +  0  =3D  2
        = ;      y =3D  2
To get z, s= et 2 for y in E4:2 + z =3D 3  à z =3D 1    
For x, set 2 for y , set 1 for z in E2:
Systems of Linear Equations
 *(-1) 
Let's elimi= nate y.
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