Mathematical Presentation of Utility maximisation:

Consumer's objective is to maximise her utility by solving UMP. To solve UMP, we set the Lagrange function of the corresponding problem, which is,    L(x₁, x₂) = U(x₁, x₂) + λ (M-p₁x₁-p₂x₂) Our objective is to maximise this Lagrange function by choosing x₁, x₂ and λ. For that we differentiate the Lagrange function by x₁, x₂ and  λ, and set all equal to zero.  

is the slope of the indifference curve and  p₁/p₂ is the slope of the budget line. So, at equilibrium we have a slope of the indifference curve that is equal to the slope of the budget line. Again, from equation (f3) we get M = p₁x₁+p₂x₂, so budget equation holds with equality sign.