1. Determine whether the system of linear equations x+y+z=3,2x-y-z=3,x-y+z=9 has No solution
Solution:
Here `x+y+z=3`
`2x-y-z=3`
`x-y+z=9`
| `|D|` | = | | `1` | `1` | `1` | | | `2` | `-1` | `-1` | | | `1` | `-1` | `1` | |
|
`=1 xx (-1 × 1 - (-1) × (-1)) -1 xx (2 × 1 - (-1) × 1) +1 xx (2 × (-1) - (-1) × 1)`
`=1 xx (-1 -1) -1 xx (2 +1) +1 xx (-2 +1)`
`=1 xx (-2) -1 xx (3) +1 xx (-1)`
`= -2 -3 -1`
`=-6`
| `|D_1|` | = | | `3` | `1` | `1` | | | `3` | `-1` | `-1` | | | `9` | `-1` | `1` | |
|
`=3 xx (-1 × 1 - (-1) × (-1)) -1 xx (3 × 1 - (-1) × 9) +1 xx (3 × (-1) - (-1) × 9)`
`=3 xx (-1 -1) -1 xx (3 +9) +1 xx (-3 +9)`
`=3 xx (-2) -1 xx (12) +1 xx (6)`
`= -6 -12 +6`
`=-12`
| `|D_2|` | = | | `1` | `3` | `1` | | | `2` | `3` | `-1` | | | `1` | `9` | `1` | |
|
`=1 xx (3 × 1 - (-1) × 9) -3 xx (2 × 1 - (-1) × 1) +1 xx (2 × 9 - 3 × 1)`
`=1 xx (3 +9) -3 xx (2 +1) +1 xx (18 -3)`
`=1 xx (12) -3 xx (3) +1 xx (15)`
`= 12 -9 +15`
`=18`
| `|D_3|` | = | | `1` | `1` | `3` | | | `2` | `-1` | `3` | | | `1` | `-1` | `9` | |
|
`=1 xx (-1 × 9 - 3 × (-1)) -1 xx (2 × 9 - 3 × 1) +3 xx (2 × (-1) - (-1) × 1)`
`=1 xx (-9 +3) -1 xx (18 -3) +3 xx (-2 +1)`
`=1 xx (-6) -1 xx (15) +3 xx (-1)`
`= -6 -15 -3`
`=-24`
Here `D=-6,D_1=-12,D_2=18,D_3=-24`
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=20 has No solution
Solution:
Here `2x+5y=16`
`4x+10y=20`
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=20` with `a_2x+b_2y+c_2=0`
we get `a_2=4,b_2=10,c_2=-20`
`a_1/a_2=(2)/(4)=1/2`
`b_1/b_2=(5)/(10)=1/2`
`c_1/c_2=(16)/(20)=4/5`
`a_1/a_2=b_1/b_2!=c_1/c_2`
So the given system has no solution (System of equation is inconsistent)
This material is intended as a summary. Use your textbook for detail explanation.
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