Linear multistep methods for initial value problems are expressed in a matrix form and used with variable order and variable step-size. The optimal order is found by a new technique that uses backward difference operator to estimate the local truncation error and find the corresponding step-size for that order. The application of such methods is equally applicable for both stiff and non-stiff systems of ordinary differential equations. The old approach represented by Gear's method is compared with our modified approach that which proved its efficiency in solving special problems. In addition, the theory and application for the joint order and step-size control in Extrapolation methods have been studied. Some new modifications is applied to decrease the error at each point of the given interval. Consequently, the global error will be less than the given tolerance.