In a three$-$compartment model, let $x_1(t),x_2(t),$ and $x_3(t)$ represent the amounts of a substance in compartments $1,2,$ and $3,$ respectively. The substance transfers between compartments as follows: $12$ with rate constant $k_{12},23$ with rate constant $k_{23},$ and $31$ with rate constant $k_{31}.$ Derive the system of first$-$order linear differential equations for $x_1(t),x_2(t),$ and $x_3(t).$