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`

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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

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