The two-dimensional linear elasticity equations are solved by the global method of approximate particular solution as a new meshless option to the conventional finite element discretization. The displacement components are approximated by a linear combination of the elasticity particular solutions and the stress tensor is obtained by differentiating the displacement expressions in terms of the particular solutions. The multiquadric radial basis function (RBF) is employed as the non-homogeneous term in the governing equation to compute the particular solutions. The cantilever beam and the infinite plate with a hole problem are solved to verify the implemented meshless method. For each situation, the trend of the root mean square error is assessed in terms of the shape parameter and the number of nodes. Unlike most of the RBF collocation strategies, it is found that numerical results are in good agreement with the analytical solutions for a wide range of shape parameter values.