A fitness center is hosting a special promotion for group classes. The center can accommodate a maximum of 200 participants. The center is offering two types of classes: yoga (Y) and spin (S). Each yoga class requires 5 participants to run, while each spin class requires 10 participants. 1. Create a two-variable linear inequality that represents the constraint of total number of participants. Clearly define your variables (5 points) 2. The center schedules 6 yoga classes and 8 spin classes. Using the inequality created in part 1 determine if this schedule meets the participant's capacity. (2 points)

Defining Variables and Creating the Inequality:

Let:

• \( Y \) = number of yoga classes scheduled
• \( S \) = number of spin classes scheduled

Given that each yoga class requires 5 participants and each spin class requires 10 participants, the total number of participants (\( P \)) can be represented by the inequality:

\[ 5Y + 10S \leq 200 \]

This inequality expresses the constraint that the total number of participants in the yoga and spin classes together must not exceed the maximum capacity of 200 participants.

Determining if the Schedule Meets Capacity:

The center schedules:

• \( Y = 6 \) (yoga classes)
• \( S = 8 \) (spin classes)

Substituting into the inequality:

\[ 5(6) + 10(8) \leq 200 \]

Calculating the left side:

\[ 5(6) = 30 \] \[ 10(8) = 80 \] \[ 30 + 80 = 110 \]

Now, we compare this to the maximum capacity:

\[ 110 \leq 200 \]

Since 110 is indeed less than 200, the schedule meets the participant's capacity.