A restaurant owner plans to use $x$ tables seating $4,y$ tables seating $6,$ and $z$ tables seating $8,$ for a total of $20$ tables. When fully occupied, the tables seat $108$ customers. If only half of the $x$ tables, half of the $y$ tables, and one$-$fourth of the $z$ tables are used, each fully occupied, then $46$ customers will be seated. Find $x.$