Afitness center has two membership plans. one has a low dollar sign up fee of 15 dollars and cost 38 dollars per month to join. a second plan higher sign up fee of 78 dollars but a lower monthly fee of 31 dollars. for what number of months is the cost the same?

Let x be the number of months.
The first plan is 15 dollars sign up fee and 38 dollars per month. So the equation is 38x + 15.
The second plan is 78 dollars as sign up fee and 31 dollars per month. So the equation is 31x + 78.

We need find when x has the same value in both equations, so we do their equality:
38x + 15 = 31x + 78
Let's subtract 15 from both sides
38x + 15 = 31x + 78
38x + 15 - 15 = 31x + 78 - 15
38x = 31x + 63

Now let's subtract 31x from both sides to have the variables on a side and the numbers on side:

38x = 31x + 63
38x - 31x = 31x - 31x + 63
7x = 63

Divide both sides by 7 to have the variable x on a side and its value on the other:
(7x)/7 = 63/7
x = 9

So at month 9, the 2 plans will cost the same.

Let's check our answers, and let y be the cost:
y = 38x + 15 = 38*9 + 15 = 357
y = 31x + 78 = 31*9 + 78 = 357
Our answer has been approved.

Hope this helps! :D