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The recent transport measurement of La$_3$Ni$_2$O$_7$ uncover a "right-triangle" shape of the superconducting dome in the pressure-temperature (P-T) phase diagram. Motivated by this, we perform theoretical first-principles studies of La$_3$Ni$_2$O$_7$ with the pressure ranging from 0 to 100 GPa. Notably, we reveal a pressure dependence of the Ni-$d_{z^2}$ electron density at the Fermi energy ($n_z^{EF}$) that highly coincides with such shape. On this basis, we further explore the electronic structure under uniaxial stress. By tracking the stress response of $n_z^{EF}$, we propose that superconductivity can be achieved by applying only about 2 GPa of compression along the c axis. The idea is further exemplified from the perspectives of lattice distortion, band structure, Fermi surface and superconducting phase coherence. We also discuss the possible charge modulation under the stress and provide an insight to the relation between n_z^EF and the superconducting Tc in La$_3$Ni$_2$O$_7$ system. Our study provides a helpful guide to the future experiment.