Determining the dimension and a basis for a vector space I have the following problem:

Let $W$ be a vector space of all solutions to these homogenous equations:
$$\begin{matrix} x &+& 2y &+& 2z &-& s &+& 3t &=& 0 \\
    x &+& 2y &+& 3z &+& s &+& t &=& 0 \\
    3x &+& 6y &+& 8z &+& s &+& 5t &=& 0\end{matrix}$$
Find the dimension of the space $W$ and determine a basis for $W$.

I tried solving the above kernel to get the solutions.
The matrix:
$$\left(\begin{matrix} 1 & 2 & 2 & -1 & 3 \\ 1 & 2 & 3 & 1 & 1 \\ 3 & 6 & 8 & 1 & 5\end{matrix}\right)$$
When performing Gauss-Jordan on it, I get the matrix rank to be $3$:
$$\left(\begin{matrix} 1 & 0 & -1 & 0 & 0 \\ 0 & 5 & 2 & 0 & 0 \\ 4 & 10 & 0 & 0 & 0\end{matrix}\right)$$
So I get lost at this point. I don't know how to get the dimension nor how to determine a basis for it.
Can anyone point out the next thing I should do and whether I started off good?

After clearing the bottom row, beneath the pivots $1$ and $5$, you'll have all zeros there. Thus you can take any $z,s,t$ and then $x$ and $y$ will be determined by $z$. Thus your basis is $\{e_1-\tfrac25e_2+e_3,e_4,e_5\}$.
