show that the equations x+2y-z=3,2x-2y+2z=2,x-y+z=-1 are consistent and solve them ?
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Explanation:
x+y+z=6,x+2y+3z=14
x+4y+7z=30
AX=D
A=
⎣
⎢
⎢
⎡
1
1
1
1
2
4
1
3
7
⎦
⎥
⎥
⎤
X=
⎣
⎢
⎢
⎡
x
y
z
⎦
⎥
⎥
⎤
Consider Aqument matrix
AD=
⎣
⎢
⎢
⎡
1
1
1
1
2
4
1
3
7
6
14
30
⎦
⎥
⎥
⎤
R
2
:R
2
−R
1
;R
3
:R
3
−R
1
AD=
⎣
⎢
⎢
⎡
1
0
0
1
1
3
1
2
6
6
8
24
⎦
⎥
⎥
⎤
R
3
:R
3
−3R
2
AD=
⎣
⎢
⎢
⎡
1
0
0
1
1
0
1
2
0
6
8
0
⎦
⎥
⎥
⎤
∴ Rank of AD=2 &
Rank of A=2
∴ It is consistent
x+y+z=6;y+z=8
x+8=6 y=k
x=−2 z=8−k
∴ The system is consistent
but has infinite many solutions
∴x=−2,y=k,z=8−k
K∈R is solution set.
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