Question

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
2 the center schedules 6 yoga classes and 8 spin classes using the inequality created in part 1 determine if this schedule meets the participants capacity

Answer 1

Let's define the variables for the problem:

- Let \( y \) represent the number of yoga classes.
- Let \( s \) represent the number of spin classes.

### 1. Create a Two-Variable Linear Inequality

Each yoga class requires 5 participants, and each spin class requires 10 participants. Thus, we can represent the total number of participants as:

\[
5y + 10s \leq 200
\]

This inequality states that the total number of participants (5 times the number of yoga classes plus 10 times the number of spin classes) must be less than or equal to the maximum capacity of 200 participants.

### 2. Determine if the Scheduled Classes Meet the Participants Capacity

The fitness center schedules:
- 6 yoga classes (\( y = 6 \))
- 8 spin classes (\( s = 8 \))

Now, we can substitute these values into our inequality to see if it is satisfied:

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

Calculating the left side:

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

Adding these together:

\[
30 + 80 = 110
\]

Now, we check the inequality:

\[
110 \leq 200
\]

Since the left side (110) is less than the right side (200), the scheduled classes do meet the participants' capacity.