solving word problems using Linear system

Question

Patrons at the Canary Club pay a one-time membership fee plus a yearly annual fee. Mark pays $8,000 after 26 years and $9,000 after (26+ 10) years. Find the membership fee and annual fee.

Answer 1

Like many word problems, we will take a common approach by first defining known and unknown information.

Known Information:

1. $8000 paid after 26 years
2. $9000 paid after 36 years

Unknown Information:

1. annual fee.
2. membership fee

Based on the problem statement, with 2 unknown quantities, we will need to have a linear system of 2 equations. A general form of this linear system is:

a₁x + b = c₁

a₂x + b = c₂

Let's assign values based on what we know and don't know:

1. The membership fee is a value that does not depend on time, therefore membership fee = b.
2. The annual fee is a value which has the units of ($/year), and affects the total payment in each of the two pieces of known information. Therefore, annual fee = x.
3. Notice that multiplying the number of years by the annual fee($/year) gives us the unit of ($), the total payment. Earlier, we noted that in 26 years, $8000 was paid, and that in 36 years, $9000 was paid.
4. Therefore, a₁ = 36, c₁ = 9000, a₂ = 26, c₂ = 8000

This brings us to putting all the information together using the system of 2 linear equations:

36x + b = 9000

26x + b = 8000

To solve this system, we can subtract the bottom equation from the top equation, and solve for x.

36x + b = 9000

- (26x + b = 8000)

______________

10x = 1000

x = 100

From there, we can solve for b, by plugging in our solved value for x into either of the original equations. This gives:

36(100) + b = 9000

3600 + b = 9000

b = 5400

Finally, we answer the question. The membership fee is $5400 and the annual fee is $100.