Joshua sponsored a fundraiser that $708$ people attended. He raised $\$ 8,640$ . He charged $\$ 15$ for balcony seats and $\$ 10$ for ground seats. How many people bought balcony seats (b) and how many bought ground seats (g)? $\newline$ $\begin{array}{c} 15 b+10 g=8,640 \\ b+g=708 \end{array}$ $\newline$ [?] balcony seats [ ] ground seats

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Q. Joshua sponsored a fundraiser that

708

people attended. He raised

\$ 8,640

. He charged

\$ 15

for balcony seats and

\$ 10

for ground seats. How many people bought balcony seats (b) and how many bought ground seats (g)?

\newline

\begin{array}{c} 15 b+10 g=8,640 \\ b+g=708 \end{array}

\newline

[?] balcony seats [ ] ground seats

Set up equations: Set up the system of equations based on the given information. $\newline$ We know that Joshua charged $\$15$ for balcony seats and $\$10$ for ground seats. The total amount raised was $\$8,640$ , and the total number of people who attended was $708$ . We can represent the number of balcony seats as $b$ and the number of ground seats as $g$ . This gives us the following system of equations: $\newline$ $1$ ) $15b + 10g = 8,640$ (total amount raised) $\newline$ $2$ ) $b + g = 708$ (total number of attendees)
Solve using elimination: Solve the system of equations using the substitution or elimination method. We will use the elimination method. $\newline$ First, we can multiply the second equation by $10$ to make the coefficients of $g$ the same in both equations: $\newline$ $10(b + g) = 10(708)$ $\newline$ This simplifies to: $\newline$ $10b + 10g = 7,080$ $\newline$ Now we have: $\newline$ $1$ ) $15b + 10g = 8,640$ $\newline$ $2$ ) $10b + 10g = 7,080$
Eliminate variable: Subtract the second equation from the first equation to eliminate $g$ . $(15b + 10g) - (10b + 10g) = 8,640 - 7,080$ This simplifies to: $5b = 1,560$
Solve for b: Solve for b by dividing both sides of the equation by $5$ . $\newline$ $\frac{5b}{5} = \frac{1,560}{5}$ $\newline$ $b = 312$ $\newline$ This means that $312$ balcony seats were sold.
Substitute and solve for $g$ : Substitute the value of $b$ back into one of the original equations to solve for $g$ . We'll use the second equation $b + g = 708$ . $\newline$ $312 + g = 708$ $\newline$ Subtract $312$ from both sides to solve for $g$ : $\newline$ $g = 708 - 312$ $\newline$ $g = 396$ $\newline$ This means that $396$ ground seats were sold.