Find the value of h,k for which the system of equations is inconsistent example ( Enter your problem )
  1. Examples
Other related methods
  1. Find the value of h,k for which the system of equations has a Unique solution
  2. Find the value of h,k for which the system of equations has Infinite solution
  3. Find the value of h,k for which the system of equations has No solution
  4. Find the value of h,k for which the system of equations is consistent
  5. Find the value of h,k for which the system of equations is inconsistent
  6. Determine whether the system of linear equations has a Unique solution
  7. Determine whether the system of linear equations has Infinite solution
  8. Determine whether the system of linear equations has No solution
  9. Determine whether the system of linear equations is consistent
  10. Determine whether the system of linear equations is inconsistent

4. Find the value of h,k for which the system of equations is consistent
(Previous method)
6. Determine whether the system of linear equations has a Unique solution
(Next method)

1. Examples





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 is inconsistent

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` × 
 `7`  `3` 
 `35`  `h` 
 `+7` × 
 `-8`  `3` 
 `-37`  `h` 
 `+6` × 
 `-8`  `7` 
 `-37`  `35` 


`=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` × 
 `7`  `3` 
 `35`  `h` 
 `+7` × 
 `6`  `3` 
 `k`  `h` 
 `+6` × 
 `6`  `7` 
 `k`  `35` 


`=(-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` × 
 `6`  `3` 
 `k`  `h` 
 `+4` × 
 `-8`  `3` 
 `-37`  `h` 
 `+6` × 
 `-8`  `6` 
 `-37`  `k` 


`=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` × 
 `7`  `6` 
 `35`  `k` 
 `+7` × 
 `-8`  `6` 
 `-37`  `k` 
 `-4` × 
 `-8`  `7` 
 `-37`  `35` 


`=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 is inconsistent

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.
Any bug, improvement, feedback then Submit Here



4. Find the value of h,k for which the system of equations is consistent
(Previous method)
6. Determine whether the system of linear equations has a Unique solution
(Next method)





Share this solution or page with your friends.


 
Copyright © 2024. All rights reserved. Terms, Privacy
 
 

.