I'm having trouble solving a linear equations system with an unknown coefficient. I am trying to solve it using the reduce row echelon form, but to no avail.

x + y + 7z = -7

2x + 3y + 17z = -16

x + 2y + (k2 + 1)z = 3k

[ 1 1 7 | -7 ]

[ 2 3 17| -16]

[ 1 2 k^{2}+1| 3k ]

Specifically, I am trying to simplify this equation to find:

a) no solution "inconsistent"

b) infinite number of solutions "consistent"

c) one solution "unique"

Right now I am just attempting a. I believe I have accidentally already solved c.

So following the procedure...

I subtract 2R_{1}fromR_{2}, giving in the second row:

[ 0 1 3 | -2 ]

I then subtractR_{1}fromR_{3}giving in the third row:

[ 0 1 k^{2}-6 | 3k+7 ]

Then subtractingR_{2}fromR_{3 }to give:

[ 0 0 k^{2}-9 | 3k+9 ]

From here I am lost. My intuition tells me that 3rd row should simplify to something like this (x representing a non-zero value):

[ 0 0 0 | x ]

Any help would be greatly appreciated.