Optimal point and MRS I read that the tangency condition is not sufficient for optimality, and that one other condition is that the MRS must equal the slope of the budget line at an interior optimum. My confusion is that since MRS is the slope of an indifference curve, then wouldn't it automatically be equal to the slope of the budget line of the budget line is tangent to the indifference curve?
The tangency condition itself is the condition that the optimal point, for an interior optimum, is the point where the MRS must equal the slope of the budget line. So the second part of your question is true. It is at the tangent point where MRS equals the slope of the budget line. You've confused yourself with the second (sufficient) condition which, on the other hand, states that the utility function must be strictly quasi-concave in order for an interior optimum to exist.
Hope this clarifies.