A health club charges a one-time initiation fee and a monthly fee. John paid 100 dollars for 2 months of membership. However, Peter paid 200 for 6 months of membership. How much will Sylvia pay for 1 year of membership?

She would have to pay $350

X = monthly fee ($)

y=one-time inititation fee

y+2x=100 (equation for John)

y+6x=200 (equation for Peter)

We have to compute the following system of equations:

y+2x=100

y+6x=200

We can solve this system by substitution systme:

y+2x=100 ⇒ y=100-2x

y+6x=200

(100-2x) + 6x=200

4x=200-100

4x=100

x=100/4=25

We find out the value of "y"

y=100-2x

y=100-2 (25)

y=100-50

y=50

one-time initiation fee is $50 and monthly fee is $25

We find out how much will Sylvia pay for a year of mebership.

sylvia will pay=y+12x

sylvia will pay=$50+12*$25=$50+$300=$350

Answer: sylvia will pay $350.