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Existence of Solution in Linear Equations
Example LN1
2x+3y=4, 3x+4y=5, x+5y=9,
6x+7y=8
⎡2
⎢3
⎢
⎢1
⎢
⎣6
3
4
5
7
4⎤
5⎥⎥
9⎥
⎥
8⎦
⎡2
⎢0
⎢
⎢0
⎢
⎣0
-1.5R1+R2→R2
-0.5R1+R3→R3
-3R1+R4→R4
4 ⎤
⎡2 3
⎢
7R2+R3→R3
0 - 0.5 - 1⎥⎥
⎢
-4R2+R4→R4
⎢0 0
0⎥
⎢
⎥
0⎦
⎣0 0
⎡2 3
⎤
⎡2 3
⎢0 - 0.5 ⎥
⎢0 - 0.5
__
⎥,
A= ⎢
A= ⎢
⎢0 0 ⎥
⎢0 0
⎢
⎥
⎢
⎣0 0 ⎦
⎣0 0
rank A=2
__
rank A =2
3 4⎤
- 0.5 -1⎥⎥
3.5 7⎥
⎥
- 2 - 4⎦
4 ⎤
- 1⎥⎥
0⎥
⎥
0⎦
n=2
__
__
rank A =3
n=2
__
rankA≠ rank A → No solution
------------------------------------------------------.
Example LN3
2x+3y=4, 6x+9y=12, 8+12y=16,
4x+6y=8
⎡2 3 4⎤
⎡2 3 4 ⎤
⎢6 9 12 ⎥
⎢0 0 0 ⎥
Row
⎢
⎥
⎥
Operations ⎢
⎢
⎢8 12 16⎥
0 0 0⎥
→
⎢
⎥
⎢
⎥
⎣4 6 8 ⎦
⎣0 0 0 ⎦
rank A=1
__
rank A =1
__
2x+3y+4z-w=-5
6x+9y+12z-3w=21
⎡2 3 4 -1 - 5⎤ ⎡2 3 4 -1 - 5 ⎤
⎢6 9 12 - 3 21⎥ → ⎢0 0 0 0 36⎥
⎣
⎦ ⎣
⎦
n=4
__
rank A =2
rankA=1
__
rankA ≠ rank A
Example LN5
→ No solution
2x+3y+4z-w=-5
4x+2y+5z-3w=-3
rank A= rank A =n → Unique Solution
Notice: Number of equations has no importance
------------------------------------------------------.
Example LN2
2x+3y=4, 3x+4y=5, x+5y=9,
6x+7y=2
4 ⎤
⎡2 3 4⎤
⎡2 3
Row
⎢3 4 5⎥
⎢0 - 0.5 - 1 ⎥
⎢
⎥ Operations ⎢
⎥
→
⎢1 5 9 ⎥
⎢0 0
- 6⎥
⎥
⎢
⎢
⎥
0 ⎦
⎣6 7 2⎦
⎣0 0
4 ⎤
⎡2 3
⎡2 3
⎤
⎢0 - 0.5 ⎥
⎢
__
0 - 0.5 - 1 ⎥⎥
⎢
⎥
⎢
A=
,
A=
⎢0 0
⎢0 0 ⎥
- 6⎥
⎢
⎥
⎢
⎥
0 ⎦
⎣0 0 ⎦
⎣0 0
rank A=2
Example LN4
n=2
rankA= rank A =r
r<n → Multiple Solution.
n-r=2-1=1. 1 variable free
Solution: Take the echolen form equation
2x+3 y=4.
Set x freely and calculate y
Set x=1 → 2 1+3 y=4 → y=2/3=0.666
Set x=0 → y=4/3=1.3333
Set x=-5.5 → y=5
⎡2 3 4 -1 - 5 ⎤
⎡2 3 4 -1 - 5⎤
⎢4 2 5 - 3 - 3⎥ → ⎢0 - 4 - 3 -1 7⎥
⎣
⎦
⎣
⎦
__
n=4
rankA=rank A =r=2
r<n multiple solution.
n-r =4-2 =2 2 variable free
To solve x and y, we use echolen form equations.
2x+3y+4z-w=-5
-4y-3z-w=7
---------------------------------Set z=0 w=1 (arbitrary selection)
2x+3y+4 0 -1=-5
-4y - 3 0 - 1=7
→ y=-2
replace y=-2 z=0 w=1 into the first equation
2x+3 (-2) -1=-5
→ x=1
Solution 1=[ x=1 y=-2 z=0 w=1 ]
------------------- ------------------Set Set z=2, w=3 (arbitrary selection)
2x+3y +4 2 - 3=-5 → 2x+3y=-10
-4y-3 2 - 3 =7
→ -4y =16 → y= -4
from first equation
2x+3y=-10 → 2x+3(-4)=-10 → x=1
Solution 2=[ x=1 y=-4 z=2 w=3 ]
---------------------------------Set Set z=0, w=0 (arbitrary selection)
2x+3y +4 0 - 0=-5 → 2x+3y=-5
-4y-3 0 - 0 =7
→ -4y =7 → y= -1.75
from first equation
2x+3y=-5 → 2x+3(-1.75)=-5 → x=0.125
Solution 3=[ x=0.125 y=-1.75 z=0 w=0 ]
Example LN6
2x+3y+4z-w=-5,
⎡2 3
⎢4 2
⎢
⎢6 1
⎢
⎢- 2 5
⎢10 3
⎣
6x+y+6z-5w =-1
10x+3y+11z -8w=-4
4x+2y+5z-3w=-3,
-2x+5y+2z+3w=-9,
4 -1 - 5 ⎤
⎡2 3 4 - 1
5 - 3 - 3 ⎥⎥
⎢0 - 4 - 3 - 1
6 - 5 -1 ⎥ → ⎢
⎢0 0 0
0
⎥
2 3 -9⎥
⎢
0
⎢0 0 0
11 - 8 - 4⎥⎦
⎢⎣ 0 0 0
0
- 5⎤
7 ⎥⎥
0⎥
⎥
0⎥
0 ⎥⎦
__
n=4
rankA=rank A =r=2
r<n multiple solution.
n-r =4-2 =2 2 variable free
Echolen form equations are
2x+3y+4z-w=-5
-4y-3z-w=7
These equations are solved in Example LN5
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