Intuition behind income and substitution effects

Okay, so you guys really don't like the income and substitution effect diagrams. I feel your pain, I really do. (see p. 467 in textbook) I want to try again to explain intuitively what is happening. You can break down the pivot of the budget line (as a result of a price change) into two steps: a slope rotation and a shift. The slope rotation is the substitution effect and the shift is the income effect. Say you start at an optimum A, as in the text, and the price of the good on the y axis decreases. For the substitution effect, you will consume more of good Y and less of good X. (everything I say is the opposite in the case of a price increase) Your utility will not change, so you will stay in the same indifference curve. How do you draw this? Find the point on the original indifference curve that has a slope equal to the new price ratio. This is your point B. Now draw a line tangent to it. This is your intermediate budget constraint. Intuitively, this is the point you would consume at if I took just enough money away from you such that the best you could do was to stay on your original indifference curve. Now, the income effect is what happens when I give you this money back. Thus it is a parallel shift of the intermediate budget constraint. (shift out with a price decrease, shift in with a price increase) To where? It shifts such that the new budget constraint touches the x axis at the same point as the old budget constraint. Now you can label an appropriate point C (depending on whether goods are normal or inferior), draw an indifference curve tangent to C, and you've found your new optimum.