Elasticity of demand functions

In the case of linear demand $d_i=a_i-x_iP$ (assuming $d_i$ is quantity demanded by individual $i$), the price elasticity of demand at point $(d_i,P)$ is begin{equation} epsilon_i(d_i,P)=x_icdot frac{P}{d_i}. end{equation} As @the_rainbox noted in their answer, price elasticity of demand varies along a linear demand curve. So in order to compare elasticities between different demand curves based only on the slope coefficients (the $x_i$'s), you need to fix $P$ and $d_i$; that is, assume that the demand curves of individuals $1$ and $2$ cross at some point $(Q_0,P_0)$. Then, you can say things like begin{equation} epsilon_1(Q_0,P_0)ge epsilon_2(Q_0,P_0) quadLeftrightarrowquad x_1ge x_2. end{equation} Or in words: $1$'s demand is more elastic than $2$'s at $(Q_0,P_0)$ if and only if $1$'s demand curve is flatter than $2$'s. [Note that since by convention demand curves are plotted in the $(Q,P)$-plane, a flat demand curve actually corresponds to a high $x_i$.]

Beware, however, that in contexts where a high degree of mathematical rigor is not required, it is sometimes taken as a rule-of-thumb that flat demand curves are "generally" more elastic than steep ones. Hence slope becomes a proxy for elasticity in those less rigorous discourses.