Slutsky Theorem - Mathematical Presentation:

We already know from the first order conditions of utility Maximisation that,  

where D_ij is the co-factor of the ith row and jth column of the determinant |D|. For income effect we know dp₁=dp₂=0, therefore we have from equation (h),  

Now for own price effect we have dM=dp2=0. So from equation (h) we get,  

Lastly, to find out own substitution effect we consider utility is constant in terms of income so, -dM+x₁dp₁+x₂dp₂=0 and dp₂=0. We have from equation