unique and infinitely solutions problem

Quote:

Originally Posted by quah13579

Consider the following system of equations, where $k\in \mathbb{R}$ is an unknown constant:

x_1−x_2+x_3 = 2
3x_1+x_2− x_3 = 2
2x_1+x_2− x_3 = k

For what values of k does the system have:
(a) a unique solution
(b) infinitely many solutions
(c) no solutions

Find the solution in cases above where the solutions exist.

please explain to me how to do it ,thanks!

Start reducing the matrix:

$\left[\begin{array}{cccc}1&-1&1&2\\3&1&-1&2\\2&1&-1&k\end{array}\right]$

After a few eliminations (I won't show out my work because matrices are tedious to type in LaTeX), I got it to this point:

$\left[\begin{array}{cccc}1&0&0&1\\0&1&-1&-1\\0&0&0&k-1\end{array}\right]$

This seems to imply that in order for there to be any solutions, $k=1$ . (If $k\neq 1$ , then our third equation is false.)

If $k=1$ , then the following equalities hold:

$x_1=1$
$x_2-x_3=-1 \implies x_3=x_2+1$

Thus $\langle x_1,x_2,x_3\rangle=\langle 1,0,1\rangle+x_2\langle 0,1,1\rangle$ .

This implies that the infinite set of solutions is spanned by the line $\langle 1,0,1\rangle+t\langle 0,1,1\rangle$ .

There seems to be no case in which there is a unique solution.