Final answer:
The y-intercept in a budget constraint scenario is the total amount of a good that can be purchased if the entire budget is spent on that good exclusively. For example, with a $56 budget and movie tickets at $7 each, José's y-intercept would be 8 movies. To find Serena's y-intercept for her gift card, additional information about the starting balance and cost per movie would be necessary.
Step-by-step explanation:
The student is asking how to determine the y-intercept of a budget constraint line from a given context. Serena received a gift card and the balance went from an initial amount to $64 after two movies and further dropped to $40 after three more movies. To find the y-intercept, we would need to know the starting balance (which would be the y-intercept) and how much each movie costs. However, since we only have information on the remaining balance, we need additional information to calculate the y-intercept. Instead, I will demonstrate how to work with similar situations with known values.
For example, if Svetlana tutors and charges a one-time fee of $25 (y-intercept), plus $15 per hour (slope), the equation for her earnings could be represented as y = 25 + 15x. The y-intercept in this linear equation represents the fixed starting fee when zero hours of tutoring are provided, and the slope indicates the rate of change in earnings for each additional hour of tutoring.
Similarly, if José has a total of $56 to spend, and the prices of goods are known ($14 for a T-shirt and $7 for a movie), we can determine the y-intercept of his budget constraint by dividing his total budget by the price of movies. This would give us the maximum amount of movies he could attend if he spent all his money on movies and none on T-shirts. Therefore, the y-intercept would be the quotient of $56 divided by $7, which is eight movies.