a movie theater has a seating capacity of 329. the theater charges $5.00 for children, $7.00 for students, and $12.00 of adults. there are half as many adults as there are children. if the total ticket sales was $ 2392, how many children, students, and adults attended?

Question

03-09-2022
Mathematics

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a movie theater has a seating capacity of 329. the theater charges $5.00 for children, $7.00 for students, and $12.00 of adults. there are half as many adults as there are children. if the total ticket sales was $ 2392, how many children, students, and adults attended?

Answer :

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CintiaSalazar CintiaSalazar · Answer 1 · 2022-09-07T13:02:36-04:00

Taking into account the definition of a system of linear equations, 178 children, 62 students and 89 adults attended to the movie theater.

System of linear equations

A system of linear equations is a set of two or more equations of the first degree, in which two or more unknowns are related.

Solving a system of equations consists of finding the value of each unknown so that all the equations of the system are satisfied. That is to say, the values of the unknowns must be found, with which when are replaced in the equations, they must give the solution proposed.

This case

In this case, a system of linear equations must be proposed taking into account that:

C is the number of children who attended the movie theater.
S is the number of students who attended the movie theater.
A is the number of adults who attended the movie theater.

On the other hand, you know:

A movie theater has a seating capacity of 329.
The theater charges $5.00 for children, $7.00 for students, and $12.00 of adults.
The total ticket sales was $ 2392.
There are half as many adults as there are children.

So, the system of equations to be solved consists of the following equations:

Equation 1: C + S + A= 329

Equation 2: 5C + 7S + 12A= 2392

Equation 3: A= C÷2= 1/2C

There are several methods to solve a system of equations, it is decided to solve it using the substitution method, which consists of clearing one of the two variables in one of the equations of the system and substituting its value in the other equation.

Replacing equation 3 in equation 1 and isolating the variable S you get:

C + S + 1/2C= 329

3/2C + S= 329

S= 329 - 3/2C

Replacing this expression and equation 3 in equation 2 you get:

5C + 7(329 - 3/2C) + 12×1/2C= 2392

Solving:

5C + 7×329 - 7×3/2C + 12×1/2C= 2392

5C + 2303 - 21/2C + 6C= 2392

5C - 21/2C + 6C= 2392 -2303

1/2C= 89

C= 89÷ 1/2

C= 178

Remembering that S= 329 - 3/2C, you get S= 329 - 3/2×178 → S= 62

Finally, substituting the value of C in Equation 3: A= 1/2C= 1/2×178 → A=89

In summary, 178 children, 62 students and 89 adults attended to the movie theater.

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