system of linear equations with parameter m

The system is as following:

$x + y - z = 1$

$x - 2 y + 2 z = m$

$3 x + y - z = 1$

I ended up getting

$x=0$

$z=y-1$

$m=-2$

How do I write that as a solution? Should I just write: $S=\{x=0, z=y-1, m=-2\}$? I feel like something is missing.

Solution 1:

Your steps are correct. There are infinitely many solutions for $(x,y,z,m)$, and all of them are of the form $(x,y,z,m)=(0,t,t-1,-2)$ where $t$ is a real number. If $m$ is already known, then the system only has a solution for $m=-2$, and the solution for $(x,y,z)$ is of the form $(x,y,z)=(0,t,t-1)$ (you can write the solution set as $S=\{(0,t,t-1)|t\in\mathbb R\}$).

Solution 2:

You end up with the line $z=y-1$ with $x=0$. There is no one solution because the system of equations is linearly dependent, meaning one equation can be written as a linear combination of the other two.