after ploting the graph for y=1.55x+30 and y=2.75X where the cost y (£) as a function of the total number for swims x for the two options (member and non member) but question is what do the gradients and y intercepts of the lines represent in terms of the situation being modelled.

To understand what the gradients and y-intercepts of the lines represent in terms of the situation being modeled, let's break down the given equations:

1. For the member option:
y = 1.55x + 30

2. For the non-member option:
y = 2.75x

In both equations, 'y' represents the cost in pounds (£) and 'x' represents the total number of swims.

Gradient:
The gradient of a line represents the rate of change between the y-values and x-values. In these equations, the gradients (1.55 and 2.75) indicate the cost increase per swim. For every additional swim, the cost increases by the corresponding gradient.

For example, if we consider the member option with a gradient of 1.55, it means that for every additional swim, the cost increases by £1.55. Similarly, for the non-member option with a gradient of 2.75, the cost increases by £2.75 for each additional swim.

Y-intercept:
The y-intercept represents the value of 'y' when 'x' is zero. In other words, it denotes the cost of membership (or non-member swim) when no swims have been taken.

In the member option equation (y = 1.55x + 30), the y-intercept is 30. Therefore, it suggests that even without any swims, there is a fixed cost of £30 for membership.

On the other hand, in the non-member option equation (y = 2.75x), there is no y-intercept term present. This indicates that for non-members, the cost starts at zero when no swims have been taken. There are no additional fixed costs involved apart from the cost per swim.

In summary, the gradients represent the cost increase per swim, while the y-intercepts represent the fixed cost component (or lack thereof) associated with the membership or non-membership options.