Simultaneous equations

Question

$x+2y-z=2 \\ x-y+z =5 \\ 3x+3y-z=\mu$

The question is for what value of the parameter $\mu$ has the system (a) no solutions, (b) one solution, and (c) infinitely many solutions.

Row reduced echelon form is

$A =\begin{pmatrix} 1& 0& 1/3& 0\\ 0 &1 &-2/3 &0\\ 0& 0& 0& 1\end{pmatrix}$

But I don't know how to use that.

EDIT: I want to know how to read the condition on $\mu$ from the matrix.

Answer 1

So we start here for no solution we just need to have just a parallel line so we just need to see for constant for 1 solution the determinant of equations should be 0 . Which proves their concurrency and for infinitely many solutions $x_1/x_2=y_1/y_2=z_1/z_2=c_1/c_2$. Thus you get the parameters for the unknown.

Answer 2

if you add twice the first equation and the second equation, you have $3x + 3y -z = 9$ and comparing with the third equation $3x + 3y - z = \mu$ have infinitely many solutions if $\mu = 9$ and has no solutions otherwise.