  We investigate the mixed-spin Blume-Capel model with spin-1/2 and spin-$S$
($S=1$, $2$, and $3$) on the simple cubic and body-centered cubic lattices with
single-ion-splitting crystal-field ($\Delta$) by using the Metropolis and the
Wang-Landau Monte Carlo methods. By numerical simulations, we prove that the
tricritical point is spin-independent for both lattices. The positions of the
tricritical point in the phase diagram are determined as
($\Delta_t/J=2.978(1)$; $k_B T_t/J=0.439(1)$) and ($\Delta_t/J=3.949(1)$; $k_B
T_t/J=0.854(1)$) for the simple cubic and the body-centered cubic lattices,
respectively. A very strong supercritical slowing down and hysteresis were
observed in the Metropolis update close to first-order transitions for
$\Delta>\Delta_t$. In addition, for both lattices we found a line of
compensation points, where the two sublattice magnetizations have the same
magnitude. We show that the compensation lines are also spin-independent.
