Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ A company that teaches self-improvement seminars is holding one of its seminars in Campbell. The company pays a flat fee of $\$208$ to rent a facility in which to hold each session. Additionally, for every attendee who registers, the company must spend $\$2$ to purchase books and supplies. Each attendee will pay $\$15$ for the seminar. Once a certain number of attendee register, the company will be breaking even. How many attendees will that take? What will be the company's total expenses and revenues? $\newline$ Once $\_\_$ attendees have registered, the company's expenses and receipts will both total $\$\_\_$ .

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Solve a system of equations using substitution: word problems

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

\newline

A company that teaches self-improvement seminars is holding one of its seminars in Campbell. The company pays a flat fee of

\$208

to rent a facility in which to hold each session. Additionally, for every attendee who registers, the company must spend

\$2

to purchase books and supplies. Each attendee will pay

\$15

for the seminar. Once a certain number of attendee register, the company will be breaking even. How many attendees will that take? What will be the company's total expenses and revenues?

\newline

Once

\_\_

attendees have registered, the company's expenses and receipts will both total

\$\_\_

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of attendees. $\newline$ The company's total expenses ( $E$ ) will be the sum of the flat fee and the cost of books and supplies per attendee. $\newline$ The company's total revenue ( $R$ ) will be the amount each attendee pays times the number of attendees.
Expenses Equation: Write the equation for the company's expenses: $\newline$ $E = \$208$ (flat fee) + $\$2x$ (cost per attendee for books and supplies) $\newline$ So, the equation for expenses is: $E = 2x + 208$
Revenue Equation: Write the equation for the company's revenue: $\newline$ $R = \$15x$ (revenue per attendee) $\newline$ So, the equation for revenue is: $R = 15x$
Set Break-even Point: To find the break-even point, set the expenses equal to the revenue: $2x + 208 = 15x$
Isolate Variable: Solve for $x$ by isolating the variable: $\newline$ Subtract $2x$ from both sides of the equation: $\newline$ $2x + 208 - 2x = 15x - 2x$ $\newline$ $208 = 13x$
Solve for x: Divide both sides by $13$ to solve for x: $\newline$ $\frac{208}{13} = \frac{13x}{13}$ $\newline$ $16 = x$
Calculate Total: Now that we have the number of attendees $x = 16$ , we can calculate the total expenses and revenues, which will be equal at the break-even point: $\newline$ $E = R = 2(16) + 208 = 32 + 208 = \$(240)$
Conclusion: Therefore, once $16$ attendees have registered, the company's expenses and receipts will both total $\$240$ .