  In this paper, we examine the universe's expansion in $ f(R, L_{m}) $ gravity
for a particular form of $ f(R, L_{m})=\frac{R}{2}+L_{m}^{n}$. The field
equations for flat FLRW metric with matter Lagrangian $ L_{m}=\rho$ are derive.
We derive Hubble parameter in terms of red-shift$(z)$ using the linear form of
Equation of State (EoS) parameter $ \omega=w_{0}+w_{1}z $. By use of $
\chi^{2}$-minimization technique and $H(z)$ datasets with $ 31 $ points, we
have obtained the best fit values of the free parameters of this model. The
evolution of the equation of state parameter$(\omega)$, energy density $ (\rho)
$, pressure $\mathit{(p)}$ cosmographic parameters, and the impact of the
energy conditions are all thoroughly examined based on the best-fit values of
the model parameters. We have also analyze Om diagnostic's behavior and observe
that the behavior of our model is quintessence. We have observed a transition
red-shift at $ z_{t}=0.736. $ We have obtain the present age of universe
$t_{0}=12.48$ Gyr for this model.
