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In the context of linear $f(\mathcal{R},T)=\mathcal{R}+χT$ gravity, where $\mathcal{R}$ is the Ricci scalar, $T$ is the trace of the energy-momentum tensor, and $χ$ is a dimensionless parameter, we have obtained exact analytical and numerical solutions for isotropic perfect-fluid spheres in hydrostatic equilibrium. Our solutions correspond to two-parametric extensions of the Tolman III (T-III) and Tolman VII (T-VII) models, in terms of the compactness $β$ and $χ$. By requiring configurations that exhibit monotonically decreasing radial profiles for both the energy density and pressure, compliance with the energy conditions, as well as subluminal speed of sound, we have constrained the parametric space of our solutions. We have also obtained analytically a parametric deformation of the T-VII solution that continuously interpolates between the T-III and T-VII models for any $χ$, and in the appropriate limits, provides an analytic approximation for the uniform density configuration in linear $f(\mathcal{R},T)$ gravity. Finally, by integrating numerically the TOV equations, we have obtained a numerical solution for the uniform-density configuration and subsequently, using the mass-radius relations, we have obtained the maximum mass that can be supported by such configurations. We have found that in the appropriate parametric regime our solution is in very good agreement with the observational bounds for the masses and radii of neutron stars.