A movie theater has a seating capacity of 319. The theater charges $5.00 for children, $7.00 for students, and $12.00 for adults. There are half as many adults as there are chil…

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asked Aug 5, 2024 78.4k views

1 Answer

KapsiR · Answer 1 · 2024-08-10T04:03:45+0000

Answer:

There were 186 children, 93 students and 40 adults attended the movie theater.

Explanation:

To solve this problem, create a system of equations.

First, define the variables:

From the information given, we know that the total number of people in the theater must equal the seating capacity:

$\implies x+y+z=319$

The number of adults is half the number of children:

$\implies z=(1)/(2)x$

The total ticket sales is the sum of the cost of each ticket type multiplied by the number of people who bought that type of ticket:

$\implies 5x+7y+12z=2326$

Now we have a system of equations:

$\begin{cases}\begin{aligned}x+y+z&=319\\z&=(1)/(2)x\\5x+7y+12z&=2326\end{aligned}\end{cases}$

Substitute the second equation into the first equation and rearrange to isolate y:

$\implies x+y+(1)/(2)x=319$

$\implies y+(3)/(2)x=319$

$\implies y=319-(3)/(2)x$

Substitute the equation for z and the equation for y into the third equation and solve for x:

$\implies 5x+7\left(319-(3)/(2)x\right)+12\left((1)/(2)x\right)=2326$

$\implies 5x+2233-(21)/(2)x+6x=2326$

$\implies (1)/(2)x+2233=2326$

$\implies (1)/(2)x=93$

$\implies x=186$

Now we have found the value of x, substitute this into the second equation and solve for z:

$\implies z=(1)/(2) (186)$

$\implies z=93$

Finally, substitute the found value of x into the equation for y and solve for y:

$\implies y=319-(3)/(2)(186)$

$\implies y=319-279$

$\implies y=40$

Therefore, the number of children, students, and adults who attended the theater was: