Solve the following system of equations:

$$
\begin{array}{rr}
2 x_{1}-2 x_{2} & =A_{1} \\
-x_{1}+4 x_{2}-3 x_{3} & =A_{2} \\
-2 x_{2}+6 x_{3}-4 x_{4} & =A_{3} \\
-3 x_{3}+8 x_{4}-5 x_{5} & =A_{4} \\
-4 x_{4}+10 x_{5} & =A_{5}
\end{array}
$$
Let's multiply each of the five equations by the numbers given below, then add them together: $(5,4,3,2,1)$; $(2,4,3,2,1)$; $(1,2,3,2,1)$; $(1,2,3,4,2)$; $(1,2,3,4,5)$.

In the first case, only the term containing $x_{1}$ remains, in the second case only the term containing $x_{2}$, and so on, so the unknowns can be directly expressed as:

$$
\begin{aligned}
& x_{1}=\frac{5 A_{1}+4 A_{2}+3 A_{3}+2 A_{4}+A_{5}}{6} \\
& x_{2}=\frac{2 A_{1}+4 A_{2}+3 A_{3}+2 A_{4}+A_{5}}{6} \\
& x_{3}=\frac{A_{1}+2 A_{2}+3 A_{3}+2 A_{4}+A_{5}}{6} \\
& x_{4}=\frac{A_{1}+2 A_{2}+3 A_{3}+4 A_{4}+2 A_{5}}{12} \\
& x_{5}=\frac{A_{1}+2 A_{2}+3 A_{3}+4 A_{4}+5 A_{5}}{30}
\end{aligned}
$$