Question

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

A company that teaches self-improvement seminars is holding one of its seminars in Greenwood. The company pays a flat fee of $876 to rent a facility in which to hold each session. Additionally, for every attendee who registers, the company must spend $8 to purchase books and supplies. Each attendee will pay $20 for the seminar. Once a certain number of attendee register, the company will be breaking even. What will be the company's total expenses and revenues? How many attendees will that take?


The company's expenses and revenues will both total how much money when
how many attendees have registered for the seminar.

Answer 1

Let's denote the number of attendees who register for the seminar as \( x \).

### Step 1: Set up the equations

1. **Expenses:**
The total expenses consist of a flat fee for renting the facility and the cost for books and supplies per attendee.
\[
\text{Expenses} = \text{Rent} + \text{Cost per attendee} \times \text{Number of attendees}
\]
Substituting the values:
\[
\text{Expenses} = 876 + 8x
\]

2. **Revenues:**
The total revenue comes from the registration fees paid by the attendees.
\[
\text{Revenue} = \text{Price per attendee} \times \text{Number of attendees}
\]
Substituting the values:
\[
\text{Revenue} = 20x
\]

### Step 2: Set the equations equal for break-even

To find the break-even point, we set total expenses equal to total revenues:
\[
876 + 8x = 20x
\]

### Step 3: Solve for \( x \)

1. Rearranging the equation:
\[
876 = 20x - 8x
\]
\[
876 = 12x
\]

2. Solving for \( x \):
\[
x = \frac{876}{12} = 73
\]

### Step 4: Calculate total expenses and total revenues

Now that we know the number of attendees is \( x = 73 \), we can calculate both total expenses and total revenues.

- **Total Expenses**:
\[
\text{Expenses} = 876 + 8(73) = 876 + 584 = 1460
\]

- **Total Revenues**:
\[
\text{Revenue} = 20(73) = 1460
\]

### Conclusion

The company's expenses and revenues will both total **$1460** when **73** attendees have registered for the seminar.

### Summary:
The company's expenses and revenues will both total **$1460** when **73** attendees have registered for the seminar.