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Use a system of equations to solve the following problem.The local theater has three types of seats for Broadway plays: main floor, balcony, and mezzanine.Main floor tickets are $52, balcony tickets are $37, and mezzanine tickets are $31. One particularnight, sales totaled $50,512. There were 86 more main floor tickets sold than balcony and mezzaninetickets combined. The number of balcony tickets sold is 139 more than 2 times the number ofmezzanine tickets sold. How many of each type of ticket were sold?

Use A System Of Equations To Solve The Following ProblemThe Local Theater Has Three Types Of Seats For Broadway Plays Main Floor Balcony And MezzanineMain Floor class=

Sagot :

Main floor ticket is $52

Balcony ticket is $37

Mezzanine ticket is $31

Let x represent main floor tickets

Let y represent balcony tickets

Let z represent main mezzanine tickets

On a particular day, total sales totaled $50512, i.e

[tex]\begin{gathered} 52\times x+37\times y+31\times z=50512 \\ 52x+37y+31z=50512\ldots(1) \end{gathered}[/tex]

There were 86 more main floor tickets sold than balcony and mezzanine tickets combined, i.e

[tex]\begin{gathered} y+z+86=x \\ x-y-z=86\ldots(2) \end{gathered}[/tex]

The number of balcony tickets sold is 139 more than 2 times the number of mezzanine tickets sold, i.e

[tex]\begin{gathered} 2z+139=y \\ y-2z=139\ldots(3) \end{gathered}[/tex]

The equatons are

[tex]\begin{gathered} 52x+37y+31z=50512\ldots(1) \\ x-y-z=86\ldots(2) \\ y-2z=139\ldots(3) \end{gathered}[/tex]

Solving to find the values of x, y and z

From equation (3), make y the subject

[tex]\begin{gathered} y-2z=139 \\ y=139+2z\ldots(4) \end{gathered}[/tex]

Substitute for y into equation (2)

[tex]\begin{gathered} x-y-z=86 \\ x-(139+2z)-z=86 \\ x-139-2z-z=86 \\ x-139-3z=86 \\ x-3z=225 \end{gathered}[/tex]

Make x the subject

[tex]\begin{gathered} x-3z=225 \\ x=225+3z\ldots(5) \end{gathered}[/tex]

Substitute for x and y into equation (1)

[tex]\begin{gathered} 52x+37y+31z=50512 \\ 52(225+3z)+37(139+2z)+31z=50512_{} \\ 11700+156z+5143+74z+31z=50512 \\ \text{Collect like terms} \\ 156z+74z+31z=50512-11700-5143 \\ 261z=33669 \\ \text{Divide both sides by 261} \\ \frac{261z}{261}=\frac{33669}{261} \\ z=129 \end{gathered}[/tex]

Substitute for z into equation (5) to find x

[tex]\begin{gathered} x=225+3z \\ x=225+3(129) \\ x=225+387 \\ x=612 \end{gathered}[/tex]

Substitute for z into equation (4) to find y

[tex]\begin{gathered} y=139+2z \\ y=139_{}+2(129) \\ y=139+258 \\ y=397 \end{gathered}[/tex]

Hence,

The number of main floor tickets (x) sold is 612

The number of balcony tickets (y) sold is 397

The number of mezzanine tickets (z) sold is 129

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