Determine values of k for which linear system of eqns has...

singing

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A linear system with three variables has augmented matrix that is row-equivalent to the
following matrix:


k + 3 2 k − 4 =3
0 2 −9 =5
0 0 k^2 + k − 2= k − 1


Determine the values of k for which the system has:
(a) exactly one solution,
(b) infinitely many solutions,
(c) no solutions.
 
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A linear system with three variables has augmented matrix that is row-equivalent to the
following matrix:


k + 3 2 k − 4 =3
0 2 −9 =5
0 0 k^2 + k − 2= k − 1


Determine the values of k for which the system has:
(a) exactly one solution,
(b) infinitely many solutions,
(c) no solutions.

The problem as posted - does not make sense. Does your row-reduced matrix look like:

k + 3 ..............2................ k − 4............=..............3
0....................2............... .....−9 ...........=..............5
0................... 0 ...... k^2 + k − 2............= .............k − 1


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Is your matrix
\(\displaystyle \begin{pmatrix}k+ 3 & 2 & k- 4 \\ 0 & 2 & -9 \\ 0 & 0 & k^2+ k- 2\end{pmatrix}\)

or
\(\displaystyle \begin{pmatrix}k+ 3 & 2k & -4 \\ 0 & 2 & -9 \\ 0 & 0 & k^2+ k- 2\end{pmatrix}\) ?

In either case, the third row corresponds to the equation \(\displaystyle (k^2+ k -2)z= k- 1\). How would you solve that equation?
For what values of k is that not possible?
 
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