Example 3 Find the values of real numbers $x$ and $y$ such that
$$
(y-1)^{2}+(x+y-3)^{2}+(2 x+y-6)^{2}
$$

reaches its minimum value.
(2001, National Junior High School Mathematics League)
Solution: Original expression $=5 x^{2}+6 x y+3 y^{2}-30 x-20 y+46$
$$
\begin{array}{l}
=5\left(x+\frac{3}{5} y-3\right)^{2}+\frac{6}{5}\left(y-\frac{5}{6}\right)^{2}+\frac{1}{6} . \\
\text { When }\left\{\begin{array}{l}
x+\frac{3}{5} y-3=0, \\
y-\frac{5}{6}=0
\end{array}\right. \text {, the above expression reaches its minimum value. }
\end{array}
$$

At this time, $x=\frac{5}{2}, y=\frac{5}{6}$, the original expression reaches its minimum value $\frac{1}{6}$.