Question

What’s the Deal?
A recreation center is offering special prices on its pool passes and gym memberships for the
summer. On the first day of the offering, a family paid $96 for 4 pool passes and 2 gym
memberships. Later that day, an individual bought a pool pass for herself, a pool pass for a friend,
and 1 gym membership. She paid $72.
1. Write a system of equations that represents the relationships between pool passes, gym
memberships, and the costs. Be sure to state what each variable represents.
2. Find the price of a pool pass and the price of a gym membership by solving the system
algebraically. Explain or show your reasoning.
3. Use graphing technology to graph the equations in the system. Make 1-2 observations
about your graphs.

Answer 1

1) Let p = price of one pool pass (dollars), g = price of one gym membership (dollars).

The two situations give the system
4p + 2g = 96
2p + g = 72

(First: 4 pool passes + 2 memberships = 96. Second: 2 pool passes + 1 membership = 72.)

2) Solve algebraically:
Divide the first equation by 2: 2p + g = 48.
Compare with the second: 2p + g = 72.

These two equalities cannot both be true: subtracting gives 0 = 24, a contradiction. Therefore the system is inconsistent and has no solution — there is no pair (p,g) that satisfies both equations. In other words, the given prices are inconsistent (either the data contain an error or the price changed).

3) Graphing the equations (for example with a graphing calculator or Desmos) you would plot
g = −2p + 48
and
g = −2p + 72.

Observations:
- The two lines are parallel (same slope −2) with different y-intercepts (48 and 72).
- They do not intersect, confirming there is no solution to the system.