Show that $u, v, w$ are in the span of $\{u+v, 2u+3v, 4v+6w\}?$ I know this has to do with linear combinations, namely that you would set out to solve the following set of equations to show that $c_{1}, c_{2}$, and  $c_{3}$ exist and are not all 0, but I'm unclear as to how I actually solve for those in this case. 
That is, I know I should have these equations:
$u = c_{1}(u+v) + c_{2}(2u+3v) + c_{3}(4v+6w)$
$v = c_{1}(u+v) + c_{2}(2u+3v) + c_{3}(4v+6w)$
$w = c_{1}(u+v) + c_{2}(2u+3v) + c_{3}(4v+6w)$
Do you not need to solve for c to do this proof?

HINT: What is the rank of the following matrix?
$$
\begin{pmatrix}
  1 & 2 & 0 \\
  1 & 3 & 4 \\
  0 & 0 & 6
\end{pmatrix}
$$
