Higher-order variational expressions are used to assess the effects of perturbing the absorption cross-section of an infinite non-multiplying homogeneous system on the response of an infinite plane detector of finite thickness placed at a certain distance from an infinite plane source. For this specific problem the number of equations to be satisfied by the weighting functions is reduced to six instead of eight. Two approaches are used to obtain these functions. One is by using Green's function approach for both the forward and backward fields and subsequently the other weighting functions. The other approach is to use trial functions provided by Monte Carlo. A methodology of utilizing virtual sampling to obtain the different weighting functions from one Monte Carlo run was demonstrated. The results revealed for this problem show that higher-order variational expressions gave more accurate estimates of the effects of perturbation than first-order expressions and the conventionally used Correlated Sampling.