Question

Lila just became a personal trainer and is finalizing her pricing plans. One plan is to charge $52 for the initial consultation and then $61 per session. Another plan is to charge $50 for the consultation and $63 per session. Lila realizes that the two plans have the same cost for a certain number of sessions. How many sessions is that? What is that cost?


For
sessions, the cost is $
on either plan.

Answer 1

Let \( x \) be the number of sessions.

For the first plan, the total cost is given by:

\[
C_1 = 52 + 61x
\]

For the second plan, the total cost is:

\[
C_2 = 50 + 63x
\]

To find out when the costs are the same, set the two equations equal to each other:

\[
52 + 61x = 50 + 63x
\]

Now, simplify and solve for \( x \):

1. Subtract \( 50 \) from both sides:

\[
2 + 61x = 63x
\]

2. Subtract \( 61x \) from both sides:

\[
2 = 63x - 61x
\]

3. This simplifies to:

\[
2 = 2x
\]

4. Divide both sides by 2:

\[
x = 1
\]

Now, we can find the cost for either plan when \( x = 1 \):

Using the first plan:

\[
C_1 = 52 + 61(1) = 52 + 61 = 113
\]

Using the second plan:

\[
C_2 = 50 + 63(1) = 50 + 63 = 113
\]

Thus, the costs are equal at 1 session, and the total cost is:

\[
\text{For } 1 \text{ session, the cost is } \$113 \text{ on either plan.}
\]

In conclusion:

For **1 session**, the cost is **$113** on either plan.