Notice the budget line will always pivot on the quantity of the goods whose price does not change. After all why would its quantity change if its price has not changed. The interesting aspect of the budget line is that it reflects what the consumer can actually buy given their income and the price of the two goods. But the consumer may be willing to buy different amounts of the goods but their budget constraint prevents them from doing so. We can show this willingness of the consumer to consume goods by constructing an indifference curve.
Lets construct an indifference curve.
Assume only two goods a and b. Assume that there are different combinations of quantities of good a and b which yields the exact same satisfaction. Each one of these combinations of A and B yield the exact same amount of satisfaction to the consumer. Now with quantity of A on the vertical axis and B on the Horizontal axis plot these different points. The picture should look like figure 2 on page 159.