[Example 5.2.3] Solve the system of equations for $x, y, z$:
$$
\left\{\begin{array}{l}
x + a y + a^{2} z = a^{3}, \\
x + b y + b^{2} z = b^{3}, \\
x + c y + c^{2} z = c^{3}.
\end{array}\right.
$$
The unknowns $x, y, z$ are from a system of three linear equations in three variables. Solving it directly is possible, but since it involves symbolic computation, the amount of calculation can be quite large. Now, by treating $a, b, c$ as the unknowns, we can view it as a cubic equation in one variable with $a, b, c$ as the roots.
Let $a, b, c$ be the roots of the equation in $t$
$$
t^{3}-z t^{2}-y t-x=0
$$

By Vieta's formulas for a polynomial of degree $n$, we have
$$
\left\{\begin{array}{l}
x=a b c, \\
y=-(a b+b c+c a), \\
z=a+b+c .
\end{array}\right.
$$

This approach makes the solution much simpler.