5. Comparing Plans with Systems

When two options each have a fixed fee plus a per-use cost, set their totals equal to find the break-even point — where they cost the same.

Worked example: Gym A charges $20 to join plus $10 per month. Gym B charges $50 to join plus $7 per month. After how many months do they cost the same?

Write each total cost as a function of months \(m\):
\[ \text{A: } 20 + 10m \qquad \text{B: } 50 + 7m \]
Set them equal:
\[ 20 + 10m = 50 + 7m \]
Subtract \(7m\): \(20 + 3m = 50\). Subtract 20: \(3m = 30\), so \(m = 10\) months.

Interpreting the answer: at 10 months both cost $120. Before 10 months, Gym A is cheaper (lower join fee); after 10 months, Gym B is cheaper (lower monthly rate). This is exactly how to choose between phone plans, streaming bundles, or rental options — compare the fixed cost against the per-use rate.