A chair $($shown below$)$ is supported in eight different ways labeled from Case #$1$ to Case #$8.$ The dimensions, loads, and coordinate system are described in Case #$1.$

Case #$1$: A fixed support at A and a fixed support at E$.$

Case #$2$: A pin at A and a fixed support at $E.$

Case #$3$: A pin at A and a pin at $E.$

Case #$4$: A roller at A and a pin at $E.$

Case #$5$: A roller at A and a roller at $E.$

Case #$6$: A roller at A$,$ a roller at $E$ and a vertical roller at $H.$

Case #$7$: A roller at A$,$ a pin at $C$ and a vertical roller at $H.$

Case #$8$: A pin at $C$ and a vertical roller at $H.$

Draw a free$-$body diagram $($FBD$)$ for each case and solve for all reactions when possible. If it is not possible, specify why.