Let's denote the number of edges that each face type contributes as $a$, $b$, and $c$, respectively. We are given that there are 4 triangles, 3 rectangles, and 1 hexagon.
Since triangles have 3 sides, 4 triangles contribute $4 \cdot 3 = 12$ edges.
Similarly, 3 rectangles contribute $3 \cdot 4 = 12$ edges.
And the hexagon contributes $1 \cdot 6 = 6$ edges.
Let the number of vertices of each face type be $x$, $y$, and $z$, respectively. We are given that there are 9 vertices in total.
Since triangles have 3 vertices, 4 triangles contribute $4 \cdot 3 = 12$ vertices.
Similarly, 3 rectangles contribute $3 \cdot 4 = 12$ vertices.
And the hexagon contributes $1 \cdot 6 = 6$ vertices.
Therefore, we have the equation $3a + 4b + 6c = 12$ for the edges, and $a + b + c = 9$ for the vertices.
To solve this system of equations, we need a third equation. Since a vertex is shared by two edges, and each edge is shared by two vertices, we have the equation $2(a + b + c) = 2 \cdot 9 = 18$.
Now we have the system of equations:
$3a + 4b + 6c = 12$
$a + b + c = 9$
$2(a + b + c) = 18$
Simplifying the third equation, we have $a + b + c = 9$, which is the second equation.
Therefore, the system of equations is:
$3a + 4b + 6c = 12$
$a + b + c = 9$
Solving this system of equations gives $a = 0, b = 6, c = 3$.
Thus, the solid has $3a + 4b + 6c = 3(0) + 4(6) + 6(3) = 0 + 24 + 18 = \boxed{42}$ edges.
A solid has faces that consist of 4 triangles, 3 rectangles, and 1 hexagon. The solid has 9 vertices. How many edges does the solid have?
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