The vector \(\vec{b}\) is in the span of the other vectors.

RREF of vectors augmented by \(\vec{b}\):

\vec b = 
\begin{bmatrix} 
 25  \\ 
 -3  \\ 
 7
\end{bmatrix}

Vectors:

\begin{bmatrix} 
 -5  \\ 
 3  \\ 
 -2
\end{bmatrix} \quad 
\begin{bmatrix} 
 -25  \\ 
 15  \\ 
 -10
\end{bmatrix} \quad 
\begin{bmatrix} 
 -5  \\ 
 -1  \\ 
 -1
\end{bmatrix} \quad 
        

\begin{solution}
The vector \vec{b}  is  in the span of the other vectors.
\[
\begin{bmatrix} 
1 & 5 & 0 & -2   \\ 
0 & 0 & 1 & -3   \\ 
0 & 0 & 0 & 0
\end{bmatrix}
\]
\end{solution}