Determine the number of vertices in a solid made up of $x$ triangles, $y$ squares and $z$ pentagons.
Without using the Euler's formula $v-e+f=2$ and without counting up all vertices by hand I am not sure how to conclude the number of vertices. I am trying to find some relationship based only on the given polygons, something like: For the number of edges I can use $$e=\frac{3x+4y+5z}{2}$$ as every edge is shared by two faces.