1. Find the value of h,k for which the system of equations 7x-7y+6z=-4,-8x+7y+3z=6,-37x+35y+hz=k has Infinite solution
Solution:
Here `7x-7y+6z=-4`
`-8x+7y+3z=6`
`-37x+35y+hz=k`
| `|D|` | = | | `7` | `-7` | `6` | | | `-8` | `7` | `3` | | | `-37` | `35` | `h` | |
|
`=7 xx (7 × h - 3 × 35) +7 xx ((-8) × h - 3 × (-37)) +6 xx ((-8) × 35 - 7 × (-37))`
`=7 xx (7h -105) +7 xx (-8h +111) +6 xx (-280 +259)`
`=7 xx (7h-105) +7 xx (-8h+111) +6 xx (-21)`
`= 49h-735 -56h+777 -126`
`=-7h-84` `->(1)`
| `|D_1|` | = | | `-4` | `-7` | `6` | | | `6` | `7` | `3` | | | `k` | `35` | `h` | |
|
`=(-4) xx (7 × h - 3 × 35) +7 xx (6 × h - 3 × k) +6 xx (6 × 35 - 7 × k)`
`=(-4) xx (7h -105) +7 xx (6h -3k) +6 xx (210 -7k)`
`=(-4) xx (7h-105) +7 xx (6h-3k) +6 xx (-7k+210)`
`= -28h+420 +42h-21k -42k+1260`
`=14h-63k+1680` `->(2)`
| `|D_2|` | = | | `7` | `-4` | `6` | | | `-8` | `6` | `3` | | | `-37` | `k` | `h` | |
|
`=7 xx (6 × h - 3 × k) +4 xx ((-8) × h - 3 × (-37)) +6 xx ((-8) × k - 6 × (-37))`
`=7 xx (6h -3k) +4 xx (-8h +111) +6 xx (-8k +222)`
`=7 xx (6h-3k) +4 xx (-8h+111) +6 xx (-8k+222)`
`= 42h-21k -32h+444 -48k+1332`
`=10h-69k+1776` `->(3)`
| `|D_3|` | = | | `7` | `-7` | `-4` | | | `-8` | `7` | `6` | | | `-37` | `35` | `k` | |
|
`=7 xx (7 × k - 6 × 35) +7 xx ((-8) × k - 6 × (-37)) -4 xx ((-8) × 35 - 7 × (-37))`
`=7 xx (7k -210) +7 xx (-8k +222) -4 xx (-280 +259)`
`=7 xx (7k-210) +7 xx (-8k+222) -4 xx (-21)`
`= 49k-1470 -56k+1554 +84`
`=-7k+168` `->(4)`
From `(1)`, we get
`=>-7h-84=0`
`=>-7h=84`
`=>h=84/(-7)`
`=>h=-12`
substitute `h=-12` in equation `(2)`, we get
`=>-63k+14*(-12)+1680=0`
`=>-63k=-14*(-12)-1680`
`=>-63k=168-1680`
`=>-63k=-1512`
`=>k=(-1512)/(-63)`
`=>k=24`
substitute `h=-12` in equation `(3)`, we get
`=>-69k+10*(-12)+1776=0`
`=>-69k=-10*(-12)-1776`
`=>-69k=120-1776`
`=>-69k=-1656`
`=>k=(-1656)/(-69)`
`=>k=24`
From `(4)`, we get
`=>-7k+168=0`
`=>-7k=-168`
`=>k=(-168)/(-7)`
`=>k=24`
The system has unique solutions if `D!=0`, so `h!=-12`
The system has infinite solution if `D=D_1=D_2=D_3=0`, so `h=-12` and `k=24`
System has no solution if `D=0` and at least one of `D_1,D_2,D_3` is nonzero, so `h=-12` and `k!=24`
2. Find the value of h,k for which the system of equations 2x+3y=5,4x+ky=10 has a Infinite solution
Solution:
Here `2x+3y=5`
`4x+ky=10`
Comparing `2x+3y=5` with `a_1x+b_1y+c_1=0`
we get `a_1=2,b_1=3,c_1=-5`
Comparing `4x+ky=10` with `a_2x+b_2y+c_2=0`
we get `a_2=4,b_2=k,c_2=-10`
For a unique solution
`a_1/a_2!=b_1/b_2`
`(2)/(4)!=(3)/(k)`
`k!=6`
For infinite solutions
`a_1/a_2=b_1/b_2=c_1/c_2`
`(2)/(4)=(3)/(k)=(5)/(10)`
`(2)/(4)=(3)/(k)`
`k=6`
For no solutions
`a_1/a_2=b_1/b_2!=c_1/c_2`
`(2)/(4)=(3)/(k)!=(5)/(10)`
There is no such value of `k`, which will satisfy the equation
This material is intended as a summary. Use your textbook for detail explanation.
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