Final answer:
The provided statement is true in the context of a proportional relationship, but if analyzing opportunity cost with budget constraints, the slope is the negative ratio of the price of goods on the axes, representing the trade-off between the goods.
Step-by-step explanation:
The statement 'For ¾, the slope is ¾ since the cost increases by 3 dollars for every 4 games Robert and his family play' is True if the context is a simple proportional relationship where the variable costs increase linearly with the number of games played. However, in the context provided about the slope of a budget constraint, which is usually represented with a negative algebraic sign, the slope would not simply be the division of the cost increment by the games played, but rather an expression of the opportunity cost of choosing one good over another.
For example, if we're talking about budget constraints and the opportunity cost between two goods, the slope would be defined as the price of the good on the horizontal axis divided by the price of the good on the vertical axis. In the provided example, the price of bus tickets is on the horizontal axis and the price of burgers on the vertical axis, making the slope $0.50/$2 = 0.25, indicating a negative opportunity cost where to purchase one good, the consumer must forego a certain amount of the other. Specifically, for Alphonso, the slope of -0.25 indicates that for every bus ticket he purchases, he must give up ¼ of a burger, or for every 4 tickets, one whole burger is foregone.