1. Determine whether the system of linear equations 2x+y+z=5,3x+5y+2z=15,2x+y+4z=8 is inconsistent
Solution:
Here `2x+y+z=5`
`3x+5y+2z=15`
`2x+y+4z=8`
| `|D|` | = | | `2` | `1` | `1` | | | `3` | `5` | `2` | | | `2` | `1` | `4` | |
|
`=2 xx (5 × 4 - 2 × 1) -1 xx (3 × 4 - 2 × 2) +1 xx (3 × 1 - 5 × 2)`
`=2 xx (20 -2) -1 xx (12 -4) +1 xx (3 -10)`
`=2 xx (18) -1 xx (8) +1 xx (-7)`
`= 36 -8 -7`
`=21`
| `|D_1|` | = | | `5` | `1` | `1` | | | `15` | `5` | `2` | | | `8` | `1` | `4` | |
|
`=5 xx (5 × 4 - 2 × 1) -1 xx (15 × 4 - 2 × 8) +1 xx (15 × 1 - 5 × 8)`
`=5 xx (20 -2) -1 xx (60 -16) +1 xx (15 -40)`
`=5 xx (18) -1 xx (44) +1 xx (-25)`
`= 90 -44 -25`
`=21`
| `|D_2|` | = | | `2` | `5` | `1` | | | `3` | `15` | `2` | | | `2` | `8` | `4` | |
|
`=2 xx (15 × 4 - 2 × 8) -5 xx (3 × 4 - 2 × 2) +1 xx (3 × 8 - 15 × 2)`
`=2 xx (60 -16) -5 xx (12 -4) +1 xx (24 -30)`
`=2 xx (44) -5 xx (8) +1 xx (-6)`
`= 88 -40 -6`
`=42`
| `|D_3|` | = | | `2` | `1` | `5` | | | `3` | `5` | `15` | | | `2` | `1` | `8` | |
|
`=2 xx (5 × 8 - 15 × 1) -1 xx (3 × 8 - 15 × 2) +5 xx (3 × 1 - 5 × 2)`
`=2 xx (40 -15) -1 xx (24 -30) +5 xx (3 -10)`
`=2 xx (25) -1 xx (-6) +5 xx (-7)`
`= 50 +6 -35`
`=21`
Here `D=21,D_1=21,D_2=42,D_3=21`
Here, `D!=0`
Hence, given system has unique solution (System of equation is consistent)
2. Determine whether the system of linear equations 2x+5y=16,4x+10y=32 is inconsistent
Solution:
Here `2x+5y=16`
`4x+10y=32`
Comparing `2x+5y=16` with `a_1x+b_1y+c_1=0`
we get `a_1=2,b_1=5,c_1=-16`
Comparing `4x+10y=32` with `a_2x+b_2y+c_2=0`
we get `a_2=4,b_2=10,c_2=-32`
`a_1/a_2=(2)/(4)=1/2`
`b_1/b_2=(5)/(10)=1/2`
`c_1/c_2=(16)/(32)=1/2`
`a_1/a_2=b_1/b_2=c_1/c_2`
So the given system has infinite solution (System of equation is consistent)
This material is intended as a summary. Use your textbook for detail explanation.
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