= Solution
Take homologous profiles and hold composition, hence <mean molecular weight>, fixed. The <stellar mass conservation equation> and <stellar hydrostatic equation> give $\rho_c\propto M/R^3$ and $P_c\propto GM^2/R^4$. With <ideal gas> pressure support, $T_c\propto M/R$. Integrating the specific nuclear rate through homologous profiles then gives
$$
L_{\rm nuc}\propto M\rho_cT_c^5\propto M^7R^{-8}.
$$
The <radiative diffusion in a star> equation gives $L_{\rm rad}\propto RT_c^4/(\kappa_c\rho_c)$. Here $\kappa_c\propto\rho_c^{1/2}T_c^{-5/2}\propto M^{-2}R$, so
$$
L_{\rm rad}\propto M^5R^{-1}.
$$
Equality of generated and transported luminosity implies $M^7R^{-8}\propto M^5R^{-1}$, whence
$$
\boxed{R\propto M^{2/7},\qquad L\propto M^{33/7}.}
$$
This is the <radiative homology with density-half inverse-five-halves opacity> scaling.
The <Stefan–Boltzmann law> defines the effective temperature by $L=4\pi R^2\sigma_{\rm SB}T_{\rm eff}^4$. Therefore $T_{\rm eff}\propto M^{29/28}$. In theoretical <Hertzsprung-Russell diagram> coordinates, \b[the slope is]
$$
\boxed{\frac{d\log L}{d\log T_{\rm eff}}=\frac{132}{29}.}
$$
The usual graphical convention puts increasing temperature to the left, so the line descends as one moves right. If the horizontal coordinate is $-\log T_{\rm eff}$, its plotted slope is $-132/29$.
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