Example 3 Mother's Day is coming, and Xiao Hong, Xiao Li, and Xiao Meng went to the flower shop to buy flowers for their mothers. Xiao Hong bought 3 roses, 7 carnations, and 1 lily, and paid 14 yuan; Xiao Li bought 4 roses, 10 carnations, and 1 lily, and paid 16 yuan; Xiao Ying bought 2 stems of each of the three types of flowers. Then she should pay $\qquad$ yuan. $[1]$
Let the unit prices of roses, carnations, and lilies be $x$ yuan, $y$ yuan, and $\sqrt{z}$ yuan, respectively. Then,
$$
\left\{\begin{array}{l}
3 x+7 y+z=14, \\
4 x+10 y+z=16 .
\end{array}\right.
$$

Eliminating $z$ gives
$$
x=2-3 y \text{. }
$$

Substituting equation (2) into equation (1) gives
$$
z=8+2 y \text{. }
$$

From equations (2) and (3), we get
$$
x+y+z=10 \text{, }
$$

which means $2(x+y+z)=20$.
Therefore, Xiaoying should pay 20 yuan.