How do I write a vector as a linear combination of other vectors. Write $\begin{pmatrix} 5  \\ 3 \\15 \end{pmatrix}$ as a linerar combination of the following vectors:  $u=\begin{pmatrix} 1  \\ 2 \\5 \end{pmatrix}$, $v=\begin{pmatrix} 3  \\ -4 \\-1 \end{pmatrix}$, $w=\begin{pmatrix} -1  \\ 1 \\1 \end{pmatrix}$.
My attempt:
$$\begin{bmatrix}
1 & 3& -1 & 5\\
2 & -4 & 1& 3\\
5&-1&1&15\\
 \end{bmatrix}\sim\to\begin{bmatrix}
1 & 0& 0 & 3\\
0 & 1 & 0 & 1\\
0&0&1&1\\
\end{bmatrix}$$
Obviously I skipped a lot of reduction steps, because it's a pain to type matrices here, but I was wondering if I even did the right thing.

So...you've done the hard part, now finish it off: 
$$c_1\begin{pmatrix} 1  \\ 2 \\5 \end{pmatrix} + c_2 \begin{pmatrix} 3  \\ -4 \\-1 \end{pmatrix}+ c_3\begin{pmatrix} -1  \\ 1 \\1 \end{pmatrix} = \begin{pmatrix} 5  \\ 3 \\15 \end{pmatrix}$$
So what have you found to be $c_1, c_2, c_3$?
Fill in the correct values for each constant, and you're done!
