Graphing a plane 
For $x\geq 0, y\leq 0, z\geq 0$
I understand how to graph each trace, and that the traces intercept each other on the coordinate axis but I do not understand why the plane is represented by the shaded region, i.e How do I know that a point in the region satisfies the equation of the plane?
 A: First trace (red)
$x-2 y=6$
has parametrization like this
$(2 t+6,t,0) $
for  $-5<t<-1$
Straight-Line is complete in x-y- plane
Second trace (green)
$x+3 z=6$
has parametrization like this
$(6 - 3 t, 0, t) $ 
for  $-1<t<1$
Straight-Line is complete in x-z- plane
Third trace (magenta)
$-2 y + 3 z = 6$
has parametrization like this
$(0,t,\frac{2 t}{3}+2) $
for  $-1<t<1$
Straight-Line is complete in y-z- plane.
Here's the picture:

Here is another one:

A: The extent to which these remarks explain anything will depend on how comfortable you are with the geometry of vector addition and scalar multiplication. :)
If $p_{1}$, $p_{2}$, and $p_{3}$ are non-collinear points of $\mathbf{R}^{n}$, and if $t_{1}$, $t_{2}$, and $t_{3}$ are real numbers whose sum is unity, then the general point on the plane through the $p_{i}$ is the convex linear combination
\begin{align*}
t_{1} p_{1} + t_{2} p_{2} + t_{3} p_{3}
  &= (1 - t_{2} - t_{3}) p_{1} + t_{2} p_{2} + t_{3} p_{3} \\
  &= p_{1} + t_{2} (p_{2} - p_{1}) + t_{3} (p_{3} - p_{1}).
\tag{1}
\end{align*}
Geometrically, this expression may be interpreted as the set of positions that can be reached by starting at $p_{1}$ and traveling by arbitrary amounts parallel to $p_{2} - p_{1}$ (the direction from $p_{1}$ to $p_{2}$) and $p_{3} - p_{1}$ (the direction from $p_{1}$ to $p_{3}$).
In your example, the axis intercepts of the plane are
$$
p_{1} = (6, 0, 0), \qquad
p_{2} = (0, -3, 0),\qquad
p_{3} = (0, 0, 2),
$$
and it's straightforward to check every point of the form (1) satisfies $2x - y + 3z = 6$. (One "smart" approach is to check that if $F(x, y, z) = 2x - y + 3z$, then
$$
F(p_{1}) = 6,\qquad
F(p_{2} - p_{1}) = 0 = F(p_{3} - p_{1}),
$$
and that "$F$ respects linear combinations" in the sense that $F(tp + q) = tF(p) + F(q)$ for all points $p$ and $q$, and all real numbers $t$.)
In case it helps, the green region (the triangle with the $p_{i}$ as vertices) is the convex hull of the $p_{i}$, namely, the set of convex linear combinations with all three "weights" non-negative: $t_{i} \geq 0$.
