Systems of equations
Y = MX +6 - line where two lines meet is the
9 + by = C -
line solution her boon equations.
if two ineslequations don't meet,
There is no solution
If bein likes are the same , where
are infinit soitions
equations in one form dx + dexz = 6 are linear.
no variables on one side
a list of all possible solutionsbo a linear system
Is called a
solution set
linear systems can
have no salvim ,
one solutions, a
infecte solutions (consistent or inconsistent)
(74x [J 2x4
eX .
X, -2xz + xz = 0 solve .
2x2 -
8x3 = 8 = 0x1 + 2x2 -
8x3 = 8
Sx1 - 5x3 = 10
, Coefficient Matrix Avamented Matrix
I· [ I
2 21
o
O
-
-
2
S0-510
X, +2 +
3
X +
2x3 =
replace equation 3 Win
-
S(291) + 293
10
=
-
Sx,+ 0x2 = 0 + Sx-St3 =
I I
1
-
2 10
293 : 1642-1043 = 10
- 8S
02
-
10-10 10
replace equation 2 won legt g
2x2-81 :8 -X-4X3 : Y
replace earation with-10leaztess
(a)
3
30X32-30 ,
X3 = -1
(xz -
4xy = 4 Yz + y =4
↳
Xz = 0
·
x
z(0) + )( ) = 0
·
,
-
X -
1 =0
,
X, =
, ElementaryNew operations
1 . replacement [replace the lowI by the Sum of itself
and of anainer
a multiple
2.
Interchange (Swa two rars)
. scaling
3 (Muriply a new by a mustifical
Two marries are few equivalent If operations
exist to transferm them from the to The sincer.
(A -
B)
is the system consistent ? (does It have a sartial
If consistent, is there I or infinite sontios ?
ex . Determine If the following system is consistent:
Xz 4xz = 8
[ii]
-
x
,
-
3xz + 2x3 = /
↑ x1 -
8x2 + 12x3 = /
Interchang row 1 and 2 to remove the zero from rowl
]
32
[ I
1 -
Chang raw 3 With row 3-yearI
01 4 g
= N Car's
-
remove the 4 In
4 -
S 12 (
T
,=
& S-e R3-wide to remove the
U In R S
Y 3x2 + 2x3 =
[1 )
-
/
:
Since we can
xz -
4x3= S
= Now selve
:
,
-
W
12 = IS Consistent
+3 system
armented has beentransformed ,
2X .
This Matrix
IS IT consistent ?
/
[, 7
2 -
32 2x
-
3xz + 2x1 =
Ixi-Uxz = S can save
0x 3 = 1S
:
Non Zero row : Any rew with at least 1 ren-Icro value
leading entry : Lefimosi non-zero value In non-zero No
Now ecuelon A Matrix
term :
is In warecocion form If
1 . any nen-zero low is above Zero raws.
2
. each leading
entry appearsIn a en to the
right of leading entry aboveli .
.
3 below a leading any
All entries in a column
are all zeros
[
O=
leadhto
J
-x X x x
000XX
000X
· o o x Ref
Y = MX +6 - line where two lines meet is the
9 + by = C -
line solution her boon equations.
if two ineslequations don't meet,
There is no solution
If bein likes are the same , where
are infinit soitions
equations in one form dx + dexz = 6 are linear.
no variables on one side
a list of all possible solutionsbo a linear system
Is called a
solution set
linear systems can
have no salvim ,
one solutions, a
infecte solutions (consistent or inconsistent)
(74x [J 2x4
eX .
X, -2xz + xz = 0 solve .
2x2 -
8x3 = 8 = 0x1 + 2x2 -
8x3 = 8
Sx1 - 5x3 = 10
, Coefficient Matrix Avamented Matrix
I· [ I
2 21
o
O
-
-
2
S0-510
X, +2 +
3
X +
2x3 =
replace equation 3 Win
-
S(291) + 293
10
=
-
Sx,+ 0x2 = 0 + Sx-St3 =
I I
1
-
2 10
293 : 1642-1043 = 10
- 8S
02
-
10-10 10
replace equation 2 won legt g
2x2-81 :8 -X-4X3 : Y
replace earation with-10leaztess
(a)
3
30X32-30 ,
X3 = -1
(xz -
4xy = 4 Yz + y =4
↳
Xz = 0
·
x
z(0) + )( ) = 0
·
,
-
X -
1 =0
,
X, =
, ElementaryNew operations
1 . replacement [replace the lowI by the Sum of itself
and of anainer
a multiple
2.
Interchange (Swa two rars)
. scaling
3 (Muriply a new by a mustifical
Two marries are few equivalent If operations
exist to transferm them from the to The sincer.
(A -
B)
is the system consistent ? (does It have a sartial
If consistent, is there I or infinite sontios ?
ex . Determine If the following system is consistent:
Xz 4xz = 8
[ii]
-
x
,
-
3xz + 2x3 = /
↑ x1 -
8x2 + 12x3 = /
Interchang row 1 and 2 to remove the zero from rowl
]
32
[ I
1 -
Chang raw 3 With row 3-yearI
01 4 g
= N Car's
-
remove the 4 In
4 -
S 12 (
T
,=
& S-e R3-wide to remove the
U In R S
Y 3x2 + 2x3 =
[1 )
-
/
:
Since we can
xz -
4x3= S
= Now selve
:
,
-
W
12 = IS Consistent
+3 system
armented has beentransformed ,
2X .
This Matrix
IS IT consistent ?
/
[, 7
2 -
32 2x
-
3xz + 2x1 =
Ixi-Uxz = S can save
0x 3 = 1S
:
Non Zero row : Any rew with at least 1 ren-Icro value
leading entry : Lefimosi non-zero value In non-zero No
Now ecuelon A Matrix
term :
is In warecocion form If
1 . any nen-zero low is above Zero raws.
2
. each leading
entry appearsIn a en to the
right of leading entry aboveli .
.
3 below a leading any
All entries in a column
are all zeros
[
O=
leadhto
J
-x X x x
000XX
000X
· o o x Ref
