[ Systems of linear equations]

Solve the system

$$
\begin{aligned}
& x_{1}+2 x_{2}+2 x_{3}+2 x_{4}+2 x_{5}=1, \\
& x_{1}+3 x_{2}+4 x_{3}+4 x_{4}+4 x_{5}=2, \\
& x_{1}+3 x_{2}+5 x_{3}+6 x_{4}+6 x_{5}=3, \\
& x_{1}+3 x_{2}+5 x_{3}+7 x_{4}+8 x_{5}=4, \\
& x_{1}+3 x_{2}+5 x_{3}+7 x_{4}+9 x_{5}=5
\end{aligned}
$$
Let's first write down the first equation, then the second equation from which the first has been subtracted, then the third equation from which the second has been subtracted, and so on:

$$
\begin{aligned}
& x_{1}+2 x_{2}+2 x_{3}+2 x_{4}+2 x_{5}=1 \\
& x_{2}+2 x_{3}+2 x_{4}+2 x_{5}=1 \\
& x_{3}+2 x_{4}+2 x_{5}=1 \\
& x_{4}+2 x_{5}=1 \\
& x_{5}=1
\end{aligned}
$$

Now we can sequentially find $x_{5}, x_{4}, x_{3}, x_{2}, x_{1}$.

## Answer

$(1,-1,1,-1,1)$.