Linear system: 3 variables, 2 equations w/o all variables The question asks to find the direction numbers for the line of intersection of the planes: 
$$ x + y + z = 1 , x + z =0 $$ 
I'm comfortable solving these sorts of linear systems when both equations include each variable. However here I'm slightly stuck. 
If I parameterize $y = t$ for instance I have:
$$x = -z$$ 
$$-z + t + z = 1$$
$$t=1$$  
From this I would guess that the directional numbers would be $(-1, 1, 1)$ (or $(1,1,-1)$ depending on if you substitute for $x$ or $z$). However the book lists that it is: $(1,0,-1)$. How did they get $0$ for $y$?

Build the augmented matrix
$$\left[\begin{array}{ccc|c} 1 & 1 & 1 & 1\\ 1 & 0 & 1 & 0\end{array}\right]$$
and then use Gauss-Jordan elimination to obtain the RREF of the augmented matrix
$$\left[\begin{array}{ccc|c} 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\end{array}\right]$$
Hence, the solution space is the line parametrized by 
$$\begin{bmatrix} x\\ y\\ z\end{bmatrix} = \begin{bmatrix} 0\\ 1\\ 0\end{bmatrix} + t \begin{bmatrix} -1\\ 0\\ 1\end{bmatrix}$$

