# Thread: For what value(s) of k does the system have....?

1. ## For what value(s) of k does the system have....?

i) For what value(s) of k does the system have;
no solutions, a unique solution, infinitely many solutions?

x + 2y - z = -3
0x + y - k-3 = -5
0x + 0y + k^2 -2k = 5k + 11

ii) each of these equations represents a plane. In each case in i) give a geometric description of the intersection of the three planes.

I have answered this already, but have gotten myself hopelessly confused, and the more I spend time on it, the worse I get. PLEASE HELP. I just can't seem to work out how to find the values of k.

2. ## Re: For what value(s) of k does the system have....?

Originally Posted by figleaf7
i) For what value(s) of k does the system have;
no solutions, a unique solution, infinitely many solutions?

x + 2y - z = -3
0x + y - k-3 = -5
0x + 0y + k^2 -2k = 5k + 11
No wonder you are confused- either you copied this incorrectly or it was given badly. I suspect it was supposed to be
$x+ 2y- z= -3$
$y- (k-3)z= -5$
$(k^2- 2k)z= 5k+ 11$

If $k^2- 2k\ne 0$ then $z= \frac{5k+ 11}{k^2- 2k}$
Now what happens if $k^2- 2k= 0$ but $5k+ 11\ne 0$? What if both are 0?

ii) each of these equations represents a plane. In each case in i) give a geometric description of the intersection of the three planes.

I have answered this already, but have gotten myself hopelessly confused, and the more I spend time on it, the worse I get. PLEASE HELP. I just can't seem to work out how to find the values of k. [/QUOTE]

3. ## Re: For what value(s) of k does the system have....?

Originally Posted by figleaf7
i) For what value(s) of k does the system have;
no solutions, a unique solution, infinitely many solutions?

x + 2y - z = -3
0x + y - k-3 = -5
0x + 0y + k^2 -2k = 5k + 11

ii) each of these equations represents a plane. In each case in i) give a geometric description of the intersection of the three planes.

I have answered this already, but have gotten myself hopelessly confused, and the more I spend time on it, the worse I get. PLEASE HELP. I just can't seem to work out how to find the values of k.
In matrix form this system is

\displaystyle \begin{align*} \left[ \begin{matrix} 1 & 2 & -1 \\ 0 & 1 & -k-3 \\ 0 & 0 & k^2 - 7k \end{matrix}\right] \left[ \begin{matrix} x \\ y\\ z \end{matrix} \right] &= \left[ \begin{matrix} -3 \\ -5 \\ \phantom{-} 11 \end{matrix} \right] \end{align*}

This matrix equation has a unique solution where \displaystyle \begin{align*} \left| \begin{matrix} 1 & 2 & -1 \\ 0 & 1 & -k-3 \\ 0 & 0 & k^2 - 7k \end{matrix} \right| \neq 0 \end{align*}.

It will either have no solution or infinite solutions where the determinant IS 0, so once you know these k values you will then need to investigate the system with those values plugged in.

4. ## Re: For what value(s) of k does the system have....?

Yes, you're right, I copied it poorly. Sorry, was on night shift.