Final Answer:
The correct equation showing a proportional relationship between x and y is option (b): y = 7x.
Step-by-step explanation:
A proportional relationship between x and y means that as one variable changes, the other changes in direct proportion to it, represented by a constant ratio. The equation y = 7x demonstrates this directly proportional relationship. In this equation, y varies with x in a consistent manner, where y is 7 times the value of x. This means that for every unit increase in x, y increases by 7 units, maintaining a constant ratio.
The equation y = 7x embodies the concept of proportionality. When x changes, y changes in a way that's directly tied to x by a factor of 7. Other options (a, c, d) don't depict a direct proportionality between x and y. For instance, option (a) represents a linear equation with a negative slope and a constant term, which doesn't indicate a proportional relationship.
Understanding the concept of proportional relationships is pivotal in recognizing equations that mirror this relationship, and among the provided choices, option (b) is the one that truly signifies a direct proportionality between x and y.