Let the ordered triples $(x,y,z)$ of complex numbers that satisfy
\begin{align*}
x + yz &= 7, \\
y + xz &= 10, \\
z + xy &= 10.
\end{align*}be $(x_1,y_1,z_1),$ $(x_2,y_2,z_2),$ $\dots,$ $(x_n,y_n,z_n).$  Find $x_1 + x_2 + \dots + x_n.$
Subtracting the equations $y + xz = 10$ and $z + xy = 10,$ we get
\[y + xz - z - xy = 0.\]Then $y - z + x(z - y) = 0,$ so $(y - z)(1 - x) = 0.$  Hence, $y = z$ or $x = 1.$

If $x = 1,$ then $yz = 6$ and $y + z = 10.$  Then by Vieta's formulas, $y$ and $z$ are the roots of $t^2 - 10t + 6 = 0.$  Thus, $x = 1$ for two ordered triples $(x,y,z).$

If $y = z,$ then
\begin{align*}
x + y^2 &= 7, \\
y + xy &= 10.
\end{align*}Squaring the second equation, we get $(x + 1)^2 y^2 = 100.$  Then $(x + 1)^2 (7 - x) = 100,$ which simplifies to $x^3 - 5x^2 - 13x + 93 = 0.$  By Vieta's formulas, the sum of the roots is 5, so the sum of all the $x_i$ is $2 + 5 = \boxed{7}.$