Example 5. The vector $\vec{d}=\{-9,2,25\}$ is to be decomposed into the vectors $\vec{a}=$ $=\{1,1,3\}, \vec{b}=\{2,-1,-6\}$ and $\vec{c}=\{5,3,-1\}$.
Solution. It is necessary to find such numbers $x, y, z$ that $x \vec{a}+y \vec{b}+z \vec{c}=\vec{d}$, i.e., $\{x, x, 3 x\}+\{2 y,-y,-6 y\}+\{5 z, 3 z,-z\}=\{-9,2,25\}$. Considering that when adding vectors, their coordinates are added and equal vectors have equal coordinates, we arrive at the system of equations

$$
\left\{\begin{array}{l}
x+2 y+5 z=-9 \\
x-y+3 z=2 \\
3 x-6 y-z=25
\end{array}\right.
$$

from which

$$
\left\{\begin{array}{l}
x=2 \\
y=-3 \\
z=-1
\end{array}\right.
$$

Answer. $\vec{d}=2 \vec{a}-3 \vec{b}-\vec{c}$.