For what values of `k` the set of equations
`2x- 3y + 6z - 5t = 3, y -4z + t=1`,
`4x-5y+8z-9t = k` has infinite solution and no solution.

Answer

`(i) kne 7 (ii) k=7`

Explanation

Given equation can be written as,
`2x- 3y+ 6z= 5t +3`
` y-4z=1-t`
`4x-5y+8z=9t+k`
which is of the form `AX = B`
Let C be the augmented matrix, then
`C=[A:B][[2,-3,6,vdots,5t+3],[0,1,-4,vdots,1-t],[4,-5,8,vdots,9t+k]]`
Applying `R_(3) rarr R_(3) - 2R_(1)`, then
`C=[[2,-3,6,vdots,5t+3],[0,1,-4,vdots,1-t],[0,-1,-4,vdots,-t+k-6]]`
`C=[[2,-3,6,vdots,5t+3],[0,1,-4,vdots,1-t],[0,0,0,vdots,k-7]]`
(i) Fpr no solution
`R_(A)neR_(C)`
`therefore kne7`
(ii) For infinite number of solutions
`R_(A)=R_(C)`
`therefore k=7`