Rodrigo just became a personal trainer and is finalizing his pricing plans. For one plan, Rodrigo charges $\$39$ for the initial consultation and then $\$42$ per session. For another plan, he charges $\$54$ for the consultation and $\$39$ per session. $\newline$ Rodrigo realizes that the two plans have the same cost for a certain number of sessions. What is that cost? How many sessions is that? $\newline$ Both plans cost $\$$ for sessions.

Full solution

Q. Rodrigo just became a personal trainer and is finalizing his pricing plans. For one plan, Rodrigo charges

\$39

for the initial consultation and then

\$42

per session. For another plan, he charges

\$54

for the consultation and

\$39

per session.

\newline

Rodrigo realizes that the two plans have the same cost for a certain number of sessions. What is that cost? How many sessions is that?

\newline

Both plans cost

\$

____ for ____ sessions.

Set up equations: Set up equations for the total cost of each plan after $x$ sessions.
Equate expressions: Equate the two expressions to find the number of sessions where costs are equal.
Simplify equation: Simplify the equation by subtracting \$$39$\ and \$$39$x\ from both sides.
Solve for x: Solve for x by dividing both sides by $3$.
Calculate total cost: Calculate the total cost for $5$ sessions using either plan.