I'm really quite confused about what *phase planes/portraits* represent. I'm only considering homogeneous systems. 

Suppose we have the system,

$$ \begin{bmatrix}x\_1' \\x\_2' \end{bmatrix} = \begin{bmatrix}1 & 2 \\3 & 2 \end{bmatrix}\begin{bmatrix}x\_1 \\x\_2 \end{bmatrix}$$

The associated phase portrait is then, 

[![enter image description here](https://i.stack.imgur.com/ldV3u.jpg)](https://i.stack.imgur.com/ldV3u.jpg)

I understand that to get the picture above, we had to take a number of points, multiply them by the coefficient matrix and plot the resulting points in the $x\_1 - x\_2$ plane. From what I've been told, the resulting phase plane is a graphical representation of $\frac{dx\_1 }{dx\_2 }$. That is what really confuses me. Why is that of any interest at all? How is that meant to give us the 'trajectories' of our solution when it doesn't contain '$t$' (assuming $x\_1$ and $x\_2$ are functions of $t$)