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A new algorithm is proposed for computing locally the linearizing output of single-input and multi-input nonlinear affine system. The algorithm modifies the extended Goursat normal form to iteratively obtain the successive integrations of one dimensional distributions of control system. The algorithm takes ideas from both vector field approach of feedback linearization and exterior differential system tools, hence the name Blended Algorithm.The proposed algorithm leads to a tower like structure depending upon the number of system inputs. Within individual tower, the coordinates are reduced one by one by finding the annihilators of vector fields at each step. The process is repeated till the single vector field is obtained for exact linearizable system. The scheme exhibits reduced computational complexity over the existing methods and can be extended to address feedback linearization of various class of control systems.