Final answer:
To find the month where the total costs of two health clubs are the same, represent each club's costs with a linear equation and solve for the number of months where the costs intersect by setting the equations equal to each other.
Step-by-step explanation:
The question relates to finding the point where the total costs of two health clubs intersect, which is a typical problem in algebra involving systems of linear equations.
To determine after how many months the total cost for each health club will be the same, one would need to establish the monthly fee and any initial costs for both health clubs. Each health club's total cost can be represented by a linear equation.
For example, if one health club has an initial joining fee of $X and a monthly fee of $Y, while the other has an initial fee of $A and a monthly fee of $B, one could set up the following equations where m represents the number of months:
Club 1: Total Cost = X + Y*m
Club 2: Total Cost = A + B*m
To find the month where the costs are the same, you would set the equations equal to one another and solve for m.
X + Y*m = A + B*m
Once the value of m is found, that will be the number of months after which the total cost for both health clubs is the same.