Three school clubs have been fundraising money for an end of the year field trip. The Science Club started out with $20 and has increased its balance by an average of $10 per week. The Music Club saved $5 per week, but started out with $50 at the beginning of the year. The Math Club was just formed this year and managed to raise $15 per week. Find all of the possible times when any two clubs will have the same money.

At any given time, the amount of money in a club's bank account will equal the starting amount plus the amount saved each week. We can set up an equation for each club that adds these two parts together to give us that total:

$ saved up by each club after X weeks = starting $ + ($ saved per week)*(X number of weeks)

For each club, here are the equations we need:

Total $ Saved by Science Club after X weeks = 20 + 10X

Total $ Saved by Music Club after X weeks = 50 + 5X

Total $ Saved by Math Club after X weeks = 0 + 15X

We can use "Y" to represent the total $ saved by each club:

Y_{sci} = 20 + 10X

Y_{mus} = 50 + 5X

Y_{math} = 15X

To find all the possible times when any two clubs will have saved the same amount of money, we just need to graph these three equations and find the points where each pair of lines intersect. After graphing the equations, we're looking for the
**X-values** for the intersection points where:

1) Y_{sci} = Y_{mus}

(The point at which 20 + 10X = 50 + 5X)

2) Y_{sci} = Y_{math}

(The point at which 20 + 10X = 15X)

Y_{mus} = Y_{math}

(The point at which 50 + 5X = 15X)

We can check the graphing solution by solving these pairs of equations with algebra. We just combine all "X" terms on one side and all number terms on the other side to solve for X.

1) Y_{sci} = Y_{mus}

20 + 10X = 50 + 5X

5X = 30

X = 6

2) Y_{sci} = Y_{math}

20 + 10X = 15X

20 = 5X

X = 4

Y_{mus} = Y_{math}

50 + 5X = 15X

50 = 10X

X = 5

Answer Summary:

1) $ saved by Science Club = $ saved by Music Club after 6 weeks;

2) $ saved by Science Club = $ saved by Math Club after 4 weeks;

3) $ saved by Music Club = $ saved by Math Club after 5 weeks.